Indeed, we have a lot of tradition, including sort of the same language, from the mother country. However, while 13 colonies did comprise the first states, we've added 37 more since then, and you'd be hard-pressed to call, for instance, the southwest U.S., from Louisiana through Texas and on to California, "British descent." You'd be just as hard-pressed to say that since much of the British isles were once a Roman colony, the current Brits are necessarily of Roman descent.
When I was growing up, the U.S. was, rightly I believe, termed "the melting pot" because of its diversity that, while honoring their many and varied traditions, remained wedded to the American ideal.
And yes, the nascent and expanding country was hard on the native Americans (who could likewise be hard on each other). But.... I'm having a tough time placing a pin anywhere on the global map where indigenous people weren't suppressed or pushed out at some point. Not to say it's right, but the U.S. is hardly standalone in that respect. (You do realize that in order to establish the colonies in the first place, the British settlers/invaders had to push out the indigenes of that time.)