Instances of 'human race' (singular) from Early English Books Online
A search of the Early English Books Online database turns up several hundred matches for "human/humane/humaine/humayne race" from the period 1576–1700. Here are the earliest ten unique instances I could find.
From a 1576 translation of A most lamentable and tragicall historie conteyning the outragious and horrible tyrannie which a Spanishe gentlewoman named Violenta executed vpon her louer Didaco, because he espoused another beyng first betrothed vnto her:
O bitch for sprong of serpents séede, / And not of humayne stocke: / But sosterd vp in desert groues, / Within some holowe rocke: / Where fostered thou waste with milke; / From ramping tygers dugge: hich thou amongst the craggy cliffes, / With sauage chappes didst tugge. / The raging senceles Lions brode, / Within themselues agree: / Detesting all against their makes, / Iniurious for to bée. / But thou deriu'd of humayne race, / Endued with reason to: / Aduenturest vpon such factes, / As they abhorre to doe.
Here the author/translator is comparing the "humayne race" with the races of serpents, tigers, and lions, where race seems to have its old meaning of (as Edward Phillips & John Kersey, The New World of Words: Or, Universal English Dictionary, sixth edition (1706) puts it, "a Lineage or Generation proceeding from Father to Son"—so "human race" effectively means "human parentage."
From Christopher Marlowe, Tamburlaine the Great (1590):
Cosroe. What means this diuelish shepheard [Tamburlaine] to aspire / With such a Giantly presumption. / To cast vp hils against the face of heauen: / And dare the force of angrie Iupiter. / But as he thrust them vnderneath the hils, / And prest out fire from their burning iawes: / So will I send this monstrous slaue to hell, / Where flames shall euer feed vpon his soule.
Meander. Some powers diuine, or els infernall, mixt / Their angry seeds at his conception: / For he was neuer sprong of humaine race, / Since with the spirit of his fearefull pride, / He dares so doubtlesly resolue of rule. /And by profession be ambitous.
Marlowe uses "humaine race" in essentially the same way that the translator of Violenta and Didaco in 1576 did.
From I. L., A true and perfecte description of a straunge monstar borne in the citty of Rome in Italy, in the yeare of our saluation. 1585. (1590):
This Pope to shew himselfe a most cruel ennemy to humane race, as most his predecessors haue been, though few in so high degree, being no sooner seated in his bloudy seat, he straight beates his braines, how to set al christendome together by the eares, partly through the pleasure he taketh in shedding innocēt bloud, partly to set himselfe the surer in his Antichristian chaire, (now almost rotten) and wholly to extinguish and roote out the glorious and moste comfortable Gospell of our Sauiour, with all the professors thereof, that hee might so bring it to passe, that their might be no memorie remaining of the one nor the other.
Here, "humane race" seems to refer to human beings as a species, and more particularly to their continuation in the face of attempts to extinguish them.
From Richard Johnson, Musarum plangores vpon the death of the right honourable, Sir Christopher Hatton, Knight, &c. (1591):
Our Cedar stock hath lost a liuely branch, / And Death the hunts-man of our humane race, / His fierce and egre appetite to stanch, / In ranging through our Forrest Syluane chace, / Hath slayne the spotles HINDE with cruell spight, / In whome his Prince reposde a chiefe delight.
Johnson treats "humane race" as a category of prey for Death, as if it were comparable to the race of deer or hares or grouse.
From Christopher Marlowe (again), Hero and Leander: begun by Christopher Marloe; and finished by George Chapman (1598):
Before them on an Altar he [the priest] presented / Both fire and water: which was first inuented, / Since to ingenerate euery humane creature, / And euery other birth produ'st by Nature, / Moysture and heate must mixe: so man and wife / For humane race must ioyne in Nuptiall life.
I read this instance of "humane race" as referring to the continuation of the line of human beings.
From Thomas Moffett, The Silkewormes, and Their Flies: liuely described in verse (1599):
Others report, there was and doth remaine / A neighbour people to the Scythian tall, / Twixt Taurus mount and Tabis fruitful plaine, / Most iust of life, of fare and diet, smal, / Louers of peace, haters of strife and gaine, / Graye ey'd, redde cheek't, and amber-headed all, / Resembling rather Gods then humane race, / Such grace appeard in words, in deeds, and face.
This is an interesting example because it compares "a neighbour people" to more run-of-the-mill "humane race" and finds that they more nearly resemble Gods.
From Thomas Floyd, The picture of a perfit common wealth describing aswell the offices of princes and inferiour magistrates ouer their subiects, as also the duties of subiects towards their gouernours (1600):
Al men are naturally borne to affect societie, whereof there be 3. sortes; the one being giuen to the engendring and procreating of humane race, as that of man and woman and this is wholy by nature, Nam omnium societatum nulta est magis secundū naturam, quam maris & foeminae. The other addicted to policie and ciuil gouernment, as lawmakers within their seueral precincts & limits; & this proceedeth partly by nature, partly by other causes. The 3. to lewdnes, and wickednes, as that of pyrates, theeues and conspirators, which societie nature needeth not: this hapneth in many places, either for want of lawe, or the execution therof.
Floyd uses "humane race" in its human lineage sense, it seems to me.
From Richard Verstegan, Odes in Imitation of the Seauen Penitential Psalmes (1601):
In later age, high God wil him abase, / And vnto low estate himself inclyne, / Mixing his nature with our humaine race, / His Godheid to our manheid to combyne: / And lo the litle lamb in strawy bed, / Shal of a maid be nowrished and fed.
...
At noontyde of our dayes wee do arryue, / As doth the Sun at midday in his height, / VVhat tyme the bel a second sound doth giue, / To moue remembrance of the heauy waight, / Of sinnes huge burthen when high heauens grace, / In humaine flesh released humaine race.
In the first instance here, "humaine race" stands in contrast to God and God's unhuman nature; in the second, it refers to humans as a (sinful yet forgiven) group of living beings.
From a 1601 translation of the works of Josephus:
He [King Antiochus] therefore commaunded these seuen [children], together with their mother Salomona who now grew well in yeeres, to bee brought before him: so they according to his commaund were brought, being of excellent beautie and worthie children of so vertuous a mother; yea they resembled angels, their faces shining like the cleare light of the Sunne, their eyes sparkled in most comely and decent maner, as testifying that they surpassed in vertue al other of humane race and condition.
"Humane race" is again equivalent to human lineage or human origin.
And from Johm Beaumont, The Metamorphosis of Tabacco (1602):
Teach me what power thee on earth did place, / What God was bounteous to the humane race, / On what occasion, and by whom it stood, / That the blest world receiu'd so great a good.
...
Then first ambitious mortals gan to rise, / And with vaine pride did the great Gods despise, / Still warr'd they with the Gods, still had the worst, / And when their hands could do no more, they curst: / Nor could the flood that inward spot deface, / Still it continued in the humane race, / Creeping vnseene, subiecting eu'ry part, / Till it possest our chiefest towre, our hart: / Which thus infected did a battell wage / Gainst the remainders of the golden age.
The use of "humane race" in this source could be read as referring to "human species," "human lineage," or both. Notably, Beaumont precedes both instances of "human race" in his poem with "the"; he is the earliest EEBO author to do this. Johnson in 1591 and Verstegan in 1601 had used the formulation "our huma[i]ne race," which may likewise be read as expressing a unitary notion of "human race." Altogether, EEBO returns 45 matches for "the human[e] race" from 1700 and earlier, and 14 more matches for "our huma[i]n[e] race" from the same period.
Instances of 'human races' (plural) in EEBO
All of the preceding instances involve "human race" as a singular term, but the EEBO database also contains three books that include instances of "humane races" as a plural form.
From a 1614 translation of Lucan's Pharsalia:
What is become of all the store / Of humane races but of yore, / That in the world were borne and bread? / The townes are vnreplenished? / The champian vninhabited? / One citie now vs all containes, / The fields manur'd by slaues in chaines.
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But thou Emathia as forlorne / Wouldst humane races hold in scorne. / As if thou wert that torrid soyle. / That Phoebus beams doth alwaies roile. / Or else that frozen ycie land / That vnderneath the pole doth stand. / Vnknowne so wouldst thou lie vnman'd.
Both of these occurrences of "humane races" arise in the context of descriptions of various peoples of what might be called the near Roman world of Latinate tribes.
From Veridicus, "O Friends! No Friends, to King, Church and State" (1648):
Now to set footing or to wade after where the unprinted Manuscript broke the ice, I desire, as the case now stands, to anchor all Fluctuations and doubts, that all the blinded eyes in the Kingdome were Swalllow-like, rubbed with Celedine or Collirium, and that all insatuated braines were purged with Hellibore, to discerne what firme friends Papists have been to Protestant Princes ever since Papisme, chiefly Jesuitisme was hatched (like all other our late renewed Heresies) from hell it selfe: and to look with in partialll eyes into their plots oft effected, if according to their sanguinnolent positions, commented by their practices in all States, they be not and ever have been as sure friends to Orthodox Princes, as Hawkes to innocent Doves, Hounds to hunted Hares, Wolves to Lambs, and all kinds of Serpents, from the Basilisk and great Regulus to the little Dipsas, to all of humane races, yet hinc nostri fundi calamitas, here is our fore-seen wrack and woe, that these Serpents that we have so long fought against to crush them, as Pigmees against Craines, ...
Veridicus casts a wider net with his "humane races," seeming to mean human lineages on a global scale.
From William Bosworth, The Chast and Lost Lovers Living Shadowed in the Person of Arcadius and Sepha (1653):
In midst of which fair Court there is a Fo[n]t, / Of Christall streams, where oft a Goddess wont, / With diverse Damsels, Goddesses I think, / Because their beauty hath such power to link / Men to their love, for sure such Heav'nly faces / Ne're sprung from mortall; ne're from humane races / But be they as they are, in that same Well / They us'd to bath, the Statues there can tell, / Chlamidias shrines th' are call'd, and strong de[f]ence / That were erected at her going thence.
As with some of the singular instances of "humane race" noted above, the point of "humane races" here is to stand in contradistinction to the races of goddesses, satyrs, and other nonhuman beings.
Conclusions
"Human race" in the sense of "human lineage" has considerable seniority over "human race" in the sociological or quasi-scientific sense of a clearly definable anthropological subgroup of human beings who uniquely share an array of specific characteristics that distinguish them from all other similarly defined subgroups of human beings. One might, therefore, reasonably ask not "Why do people talk about 'the human race' when there are so obviously multiple human races?" but rather "Why did people start talking about multiple 'human races' when English speakers were already referring to 'the humane race' as a single grouping that encompassed all human beings, at least as early as John Beaumont in 1602?"
Today, we have the quasi-scientific language of "human races" (which has antecedents in very old references to ethnically similar lineages or families of people from adjacent or nearby regions), and we have the 400-year-old (or older) notion of "the human race" as a collective term for all human beings. If usage determines validity, neither of these terms is illegitimate as a popular (that is, nonscientific) way to conveniently divide or unite people.