If you’re eligible for need-based aid, then that will change the picture. Have you run Net Price Calculators for some of your full-need-met private U’s? Start with Cornell: https://npc.collegeboard.org/student/app/cornell Also see what you get with Rice and Mudd. All three meet full need but the formulas may differ. And even two schools that meet need and show a similar EFC may differ in the extent to which loans are a part of the aid package.
(Note that unless it is one of the most generous “no loan” schools, your daughter’s financial aid package will likely include loans already. So when you propose taking out additional loans to bridge the gap between your EFC and what you can actually afford… that will be additional debt beyond what’s built into her aid. The amount your daughter can borrow in her own name will be limited. Think very, very seriously about your own financial future before taking on parent loans.)
If the NPC produces a number you can afford, then apply away! It’s not a slam-dunk for any of the reach schools but certainly she has a shot.
Likewise, run the NPC for every school you’re considering and see if the need-based aid gets you to your price point. Those OOS publics are not going to give financial aid, though, so you have to figure on the “sticker price” minus any merit… which means my previous statement that they’re not affordable likely stands.
If full-need-met schools work for you, then there are more good possibilities to add, that I didn’t suggest previously. Tufts, USC (as you mentioned), Vanderbilt, Duke, UPenn (but so ED-heavy that it’s almost not worth it to apply RD), Northwestern, WashU, etc.
Some of the merit aid schools I suggested before, like Rochester, CWRU, Northeastern and Tulane, also meet or nearly-meet full documented need. The two types of aid will not “stack,” however - the only way you’ll pay less than the EFC produced by the net price calculator is if the merit scholarship by itself is bigger than your documented need. If you were to get, say, 20K in need-based aid and 20K in merit aid, the merit aid would simply replace the need-based aid and your out-of-pocket would stay the same.
Carnegie Mellon is stingy on aid. You can see what your EFC is there, but they generally do not meet full documented need.
The Wikipedia page on Need-Blind Admissions actually has pretty good information on this topic. Need-blind really does mean that whether you apply for aid is not part of the admissions decision. Even among schools that are need-aware, many do not consider financial need on the first pass - they use a need-blind process to identify the first batch of acceptances, and then favor full-pay applicants in filling the rest of the spots, so as not to exceed their financial aid budget. Look at both the NPC and whatever info is available on the admissions practices of a particular need-aware school. If you were only eligible for a small amount of aid anyway, or if it’s a generous-merit school and you’re willing to bank on merit aid, then you might think about the strategy you suggested of not applying for aid; but if there’s significant financial aid at stake, it’s really better to apply.
Hope that helps! See what the NPC’s say and then you’ll have a lot more clarity on what your strategy should be.
P.S. Michigan State is another terrific STEM school with merit potential, that I should have mentioned earlier.