They'd like to go to top universities because they're the best. Being private is just one more filter that stops poor people from getting the education they deserve.
It's more about capacity. There are hundreds more public universities than private. The money picture is ... messy and more complex. Some private universities are cheaper for low-income students than public schools (Harvard), some are drastically more expensive.
And not all private universities are "better" than all public universities. Think Berkeley, UT, umich, etc. All fantastic and in many areas clobber many / most private universities. There are some obvious outliers at the very top-ranked schools but, statistically, almost nobody goes to them.
It's messy and complex, but at the end of the day, the public universities educate drastically more people and generally do a good job of it; they have to be considered any time we talk about access to education.
Quality education can be public and free.