I think the vast majority of Americans would not object to the government building solidarity and shared values. That is and always has been a core function of the educational system. The debates, to date, have raged about what those values should be.

My pet theory is not just that homogeneity of culture makes it easier to share wealth, but that it makes most things easier: dealing with crime, dealing with education, etc. E.g. in Chicago, there is a deep schism between the heavily black south side, and the whiter north side. Now, every city has poor parts and rich parts, but what's really stark about Chicago is how completely the two parts disassociate from each other, due to in no small part a lack of shared culture.[1] This is, of course, to the detriment of both. The school system (CPS) is a primary example. Its a deeply dysfunctional piece of welfare: generous on one hand (relatively high spending/student), but crippled by the fact that whites and the middle/upper classes all but "opt out" of the system (CPS is 90% minority and 90% low-income).

[1] My wife grew up in a small town in Iowa, where there were also rich people (doctors, store owners) and poor people. But there was tremendous shared culture: everyone went to the same high school, one of a small number of churches, participated in the same set of after-school activities, etc.