Should I apply for scholarships?

Answered

So this can be complicated and I'll try to break it down as best I can.

First, yes, SOME private schools will cover most of your tuition if your family makes less than $60k/yr. This is most true for top colleges—think Ivy League schools, Duke, Northwestern, Stanford, Vanderbilt, Rice, MIT, etc. but it's not just limited to them. Some smaller private universities and liberal arts colleges will do the same.

Regarding those top schools, you have the grades and academic chops to get accepted to those schools, you would likely be able to go to one of them for nearly free. Some of them will include federal student loans in their aid packages (this is called "self-help" aid), but those typically aren't more than $5000/yr or so. Several guarantee to only offer grant aid (money you don't have to pay back)—which is basically all of the Ivies except Cornell. So those schools would be by far your cheapest options.

However, not all private colleges will do this. What you need to look for when you're searching for schools is schools that say they will meet "full demonstrated financial need" or "100% of demonstrated financial need." For instance, you can see on the financial aid page for the University of Richmond here: https://financialaid.richmond.edu/prospective/need-based/index.html that they meet "100% of our students' demonstrated need." If a school says that, then that means they would more than likely cover most of your cost of attendance with financial aid. The exact amount they give you can change from year to year depending on your family's financial circumstances, but you can get an approximate idea of what you might get from a specific school by checking their net price calculator (for example, here's Richmond's: https://financialaid.richmond.edu/prospective/need-based/cost-calculators.html).

HOWEVER, if a school doesn't say that they'll meet full demonstrated need/100% of financial need/etc., they will probably not cover most of your tuition and cost with grant money, and imo are not worth applying to. A big example of this is NYU, which despite being a very good university does not offer very good financial aid. So when you apply, make sure you check for that language on their financial aid page as you decide on where to apply.

With all of that in mind, applying for third-party scholarships on places like Chegg or CollegeBoard typically would not be worthwhile for someone in your situation. You're correct that colleges will typically just offset your need-based aid with that money, so you would have to earn more in those scholarships than your total need-based package to make them worthwhile. Since most scholarships are very small, that is highly, highly unlikely, and the time spent applying for them is likely better spent working a job (you'd get work experience for your resume and make actual money). You can still try for very large scholarships, but the smaller ones—$500, $1000, etc. a piece—are probably not worth it.

I hope that all makes sense. Feel free to comment if you have follow-up questions.