The expected family contribution for a low SES student at Harvard is around $2 to $4K a year, so the above student is much more the exception than the norm.
At the end of the day, cast a wide net that includes 2 or 3 Florida publics, and a few out of state private and public universities. When looking at OOS or private schools, you’re looking for scholarships that pay at least full tuition (otherwise Florida Public’s with Bright Futures and likely merit scholarships will prove more affordable).
Many of these scholarships are competitive (Like UF"s MFOS or UM’s Hammond’s), so you’ll have to be accepted and then wait to see if you qualify for one of these scholarships, before making a finale choice.
You can find a lot of information in the Financial Aid and Scholarships forum.
Good Luck!