Why did Americans come up with new words for the British English words like "sidewalk" instead of "pavement"?
I only see a reason they omitted certain letters in some words, like "colour" - "color", but I don't see any reason why they would make their own words.
Why didn't they stick with the British words?
It would be impossible to supply a general answer to this. It is a vast topic, and potentially the subject of a course of study.
Many years ago I read Mother Tongue by the Anglophile American, Bill Bryson, and if you are seriously interested in the matter I would recommend you to Bryson's work. He does go into it at some length.
However in answer to your specific question about "sidewalk", as I'm sure you realise the equivalent in Britain is "pavement".
Now "pavement" has existed since at least the 14th century as a paved area (not necessarily a footpath at the side of a road):
Sense 1a. A paved surface; a hard covering laid on the ground, outside or (less commonly) in a building, formed of stones, bricks, tiles, or similar materials fitted closely together, usually on a prepared bed of hard core; (also) a similar covering formed of a layer of cement, concrete, asphalt, etc. Now chiefly in technical contexts. In early use also occasionally: †the hard ground (obs.). (OED).
It did not specifically become associated with a raised path, next to a road, until the early 19th century. Jane Austen in Pride & Prejudice (1813) says:
Kitty and Lydia..led the way across the street..and fortunately had just gained the pavement when the two gentlemen turning back had reached the same spot.
Meanwhile in America, the word for the metalled part of the highway tended to be "pavement", which had been the case in Britain too, and still is in engineering circles.
And in the 18th century the Americans had adopted the word "sidewalk", which was nothing new. It is evident in Britain from the early 17th century.
1605 in W. M. Metcalfe Charters & Documents Burgh of Paisley (1902) 279 The utter sydewalk nuik of the bakehous.
1674 Dryden & W. Davenant Shakespeare's Tempest (new ed.) i. ii. 5
'Tis composed of three Walks of Cypress-trees, each Side-walk leads to a Cave... The Middle-Walk is of a great depth.1718 S. Switzer Ichnographia rustica II. 197 A few of these Walks are absolutely necessary, in Respect to the Grandeur..of a Situation, as the Middle and Side Walk.
All above quotations from OED online
