Edit: As another poster pointed out, you may not be familiar with how the term “financial aid” is used in the US. It is used as a generic term for any sort of financial support provided by a college that helps reduce your cost of attending. So, a grant you get and do not have to pay back, also known as a scholarship, is a form of financial aid. And colleges can give you grants based either on need, or merit, or a combination of both. All of that would be included in the definition of financial aid. Unedit.
The US colleges that have need-based aid for Internationals are definitely not all within that list of 20 research universities. Just to begin with, there are something like 70 (give or take) colleges that are what are called meets need for Internationals, which means they will offer enough aid to cover all of what they calculate to be your need. And then there are more that do not guarantee that, but have some budget for International need aid.
In general, for practical purposes it really should not matter to you whether a college calls its International aid need-aid, merit-aid, or a combination. So rather than try to micromanage how the college offers aid, I would just look for any college that makes any offers as large as you need, and has MechE.
Speaking of which . . .
So one thing you should get used to doing is checking a college’s CDS, Section H6, to get an idea of what is really happening.
Like, let’s look at UTK:
They apparently gave out 48 aid awards total to Internationals. I believe this is only about 1 in 5 Internationals at UTK, so most don’t get aid at all.
OK, then the average award was $11,137. Tuition and fees for non-TN-residents is over $33K. So even these few were mostly not getting anything close to full tuition.
And in fact, I believe this is the program you might have had in mind:
The award ranges from $8K to $18K annually. That is consistent with their reported average. But even the maximum award of $18K is not full tuition and fees. It is over $15K short of that.
So based on what you have said, these sorts of colleges are not going to work for you, because their maximum aid is not enough to come close to a full tuition award.
Again, I realize all this may not be good news, but I think you need to understand all this. The number of colleges in the US with a MechE program and aid for Internationals–need, merit, or need+merit–that gets as high as covering all tuitions and fees is very low. And it doesn’t include colleges like UTK (as far as I can tell at least).