That would assume the free education people get is equivalent across income levels, which is a hard thing to do in the best of circumstances.

Even independent of educational quality, there is a world of difference between a school's ability to teach a well fed, comfortable child and a child whose only meal of the day is the free school lunch. Similarly, a rich child can afford to do unpaid internships on summer break, whereas a poor one may be working minimum wage jobs to generate an income instead.

This is before you get into the weeds of things like teachers preferring to be placed into good schools because it makes their jobs easier, that well-connected schools can raise money outside of the public system, etc. People say they want educational equality, but this is almost never at the perceived cost to their own children.