Harvard isn't a good example, they offer tremendous financial aid. If your household income is less than $65,000 (which could count as lower middle class), you pay nothing. By their figures 90% of households in America would pay less for Harvard than for a state school. [0]

There are many private colleges with a higher "sticker price" than Harvard but with a much lower discount rate (i.e. what the average student pays is much closer to that sticker price).

[0] https://college.harvard.edu/financial-aid/how-aid-works/cost...