As I perceive this question, it could be considered as a problem in the semantics of English, in a totally virgin area of the linguistics of this language that would be language planning, in the field of sociolinguistics. Technically, and informally, this is not an absolutely void area of activity, and the question in the OP becomes acceptable, provided it is taken as in the OP's text, that is, as "Why not just use 'animals' to indicate (factual)'animals'(including humans)?". I propose an answer because I happen to consider as self-evident the facts in favour of not doing so; I use otherwise the support of no known authotity on the subject.
As highlighted by user Stuart F, "the Judaeo-Christian creation myth clearly separates humans from other animals.", and this fact, if not exclusive in giving rise to a clear separation between the two in the thinking at the foundation of Western culture (therefore Anglo-Saxon culture), reflects a corresponding state of affairs in the present day secular culture in those realms. That is to say, this separation exists as a principle that operates beyond the bounds of strict religious culture, and that is so in spite of the waning influence of that latter (which raises questions as to the part it has played in this separation, but that is not the point). The distinction is therefore as important and useful as ever; preserving it allows an uncomplicated access to the literature, past and present. Moreover, it is not detrimental to modern day changing attitudes towards animals in the way of preserving and advocating more humane practices as regards to how, we, humans, treat them; after all it is nowadays thought to be well to delineate clearly gender differences.
As again highlighted by user Stuart F, "historically, the idea that humans are animals has only been widely accepted from the late 19th century", and it can be said, I believe, that this new scientific reality has still not really displaced traditional thinking; we turn a rather favourable eye towards the idea of using animal tissue in medicine, but animals, more than ever, cannot attain to any equal position with humans: in the choice of whom shall perish, of the family pet and a child in this same family, there is no dilemma, in the choice of whom shall be eaten first in a case of perduring siege, there is (bar exceptions) no dilemma, etc. It is a fact that only certain minority currents of spiritual thinking place human beings on a higher plane of existence, yet human life is still quasi-generally deemed of much greater value, and we generally think this should remain so. It appears then that the impact of scientific progress leaves traditional values on the whole unaffected.
The argument just proposed aims at making salient the importance of the distinction embodied in the word "animal" in its traditional acceptation, and therefore, at stressing that its definition shouldn't be changed. In the eventuality of this being the chosen manner of proceeding, we are left with the havoc "created" by scientific progress, which tells us that animals are biological entities very similar to human beings, that, therefore, those latter can be considered to be also animals, and that, as from now, they will be so called. But this late introduction of a new definition into the language is nothing more than a fact of language planning that has been left solely to specialists in the domain of zoology! Since when do we know of perfect, impeccable planning that never had to be modified and adjusted? It is only needed to consider in the field of astronomy the recent case of scientists having to contend with such an instance of erroneous evaluation, and to revise their outdated thinking, to in turn make known that what had been considered a planet for quite some time was not in fact a planet. In the encyclopedias, the description of the celestial body known as Pluto will have to be changed from "planet" to "dwarf planet". This is nothing exceptional in the world of science.
Is Pluto a planet?
And then, in 2006, the International Astronomical Union (IAU) reconsidered Pluto’s status. In a controversial vote, astronomers — not planetary scientists — “demoted” Pluto to the status of being classified as a dwarf planet, taking away one major planet and reducing the number in our solar system to eight. Astronomers suddenly took sides, seeing various sides in the logic, and schoolchildren all around the world were heartbroken, having been enamored with the story of the most distant and mysterious planet that was discovered by a young, self-educated researcher, and having that status heartlessly yanked away.
[…]
For many years prior to Pluto’s reclassification, astronomers had come to realize that the Kuiper Belt, the cloud of small bodies in the outer solar system, holds countless thousands of icy objects, and some of them are relatively large. Would astronomers eventually have a situation in which they had many more planets to add to the equation? Nerves began to be rattled. Pluto, some reasoned, could be the tip of the iceberg of an entirely new class of countless objects.
What makes the community of zoological scientists immune to the possibility of having to revert to revisionist practices? The path to follow is clear: let the newfangled and brash decision be reconsidered and make place to a new term in the scientific field, a term that truly reflects the scientific reality and that does not imply tampering with solid tradition. What keeps that body of scientists to decrete (possibly assisted by linguists or the linguistically minded among them) that from now on animals and human beings are to classified as belonging to the kingdom of, for instance, bestions? Of course, this is a mere example, and plenty of possibilities can be examined so as to try to comply best with a variety of criteria. Do we have to be reminded every time we are confronted with the concept "animal" as meaning "non-human", of the fact that we, humans, are nothing more than something like animals? I don't think so.