OP, you’re in a difficult spot. It sounds like you come from a family that has some means. You describe yourself as someone who will not qualify for need-based aid. This means that you will be full pay or close to it.
If you’re family’s limit/willingness to pay is 35K, then you have options, but none of them will be (though I’m open to hear others correct me) Princeton/MIT-level. Your first option is to target LACs (liberal arts colleges) that offer merit aid. The elite LACs are out of the running (Amherst, Williams, Bowdoin, Dickinson, Swarthmore, Vassar, and on, and on, and on) because they will not offer merit, or if they do, the amount will not be in the range you are looking for. For instance, Skidmore offers a few (or maybe just one?) merit award, but nothing that would lower the sticker price of the college to 35K. You can Google to find a list of the northeast schools (a combination of private universities and LACs) that do not offer merit award at all. Again, even the LACs that do (St. Lawrence, Wheaton, maybe Washington and Lee) are not going to offer the levels of merit award that you are looking for. Plus, they do not meet your “perceived” quality level.
Families in your situation do have threads here on CC, so search around. I remember one thread where a parent was in your exact situation. She wanted to send her child (a son, I believe) to a high quality school but had a firm cap of 35K per year. Her solution–and this is a common strategy–was to target non-elite schools (though still highly regarded, even prestigious in their own ways) outside the NE. But this means being open to schools in the Midwest and South. Pennsylvania (your state) is a mixture, as it contains some elite colleges where merit money is either not given (Swarthmore, Haverford) or will not be game-changing, but it also has colleges that do offer substantial merit, essentially to lower the sticker price of the college and attract students like you (Allegheny, Ursinus, Juniata). The CTCL schools (the three I just listed are CTCL schools) are popular options, as they all (as far as I know) offer merit money. These also include schools like Beloit, Earlham, Rhodes, Knox, Southwestern, Lawrence, Kalamazoo. Schools outside the NE are often already 10K less than the elite schools out east, and if the student can earn 20K in merit, that essentially makes the college 30K cheaper than the more expensive schools in the NE.
But I’m sure you see the bind that you’re in. These schools, though excellent (note: I am biased, as I attended a midwestern CTCL LAC), do not have the same name recognition as the elite LACs, let alone the large, prestigious private universities (Ivies, Stanford, Chicago, NW, Vandy, Wash U). One has to sort of lose the prestige-obsession and be confident that there are many fine colleges–that will offer merit money–that are well-regarded and are top-notch but are often overlooked or fly under the radar. The other challenge is that you’ll have to be a very strong student (though your basic stats look excellent) to earn significant merit. Significant merit might be half-tuition. My alma mater offers five full-tuition scholarships to incoming students, but five is a small number, of course. Those five students who receive full tuition (reducing the cost of the college to room and board and fees will essentially be very high-achieving scholars. Not all of these schools will have full-tuition scholarships, and even if they do, you cannot assume that you’ll receive one.
The other option is to target in-state public schools. You’re from PA, so you might see what sorts of financial numbers Penn State generates. With this country’s current economic challenges, combined with many state budget cuts affecting education, public universities do not always have a lot of money to throw around. Of course, Penn State, a fine school, does not meet your Princeton/MIT-desired level. I am sorry to say, unless someone else here posts a different route, that such schools will not be possible, not at 35K per year for someone whose family will be full pay (or close to it).