Very very few US colleges have an international student population of more than 15%; most are in the 8-15% range, whether public or private, large or small. You find more international students than you might expect at large state universities: for the same reason that people are suggesting them to you: the costs are often lower, in part because they are often in places with a lower cost of living, and in part because the incremental cost to the university of each additional student is relatively small.
There are challenges to being a non-national student anywhere in the world: most of your classmates are nationals, and while everybody is learning about going to school/university together, and learning the academic material together, you are also learning how to live in that culture and adapt to the local way of doing things.
It takes commitment on the part of the non-national student to develop real friendships- people will generally be friendly, but it is definitely the outsider who has to make more of the effort: to keep putting themselves out there, figuring out the social cues, not over-weighting what might or might not be deliberate slights, etc.
Thanks to frequent international moves, we have done this many, many times in our family, I have seen it and/or experienced it at every level from pre-school to doctoral programs, across 4 continents. It will be hard, whether you are at a small private university or a large public university, and there’s no point pretending otherwise. IMO, it’s worth it- but it’s important to go in with realistic expectations.