My son didn’t choose the cheapest school he got into, although it made it to his final three. I’m suggesting you do two things. 1) decide what you want out of a school based on more than just the ranking. 2) consider the real cost of spending more, in your case LOTS more. College is partly investment and partly experience. The experience component is important, but doesn’t have a tangible ROI.
Neither Mudd or Cornell are known as nurturing schools.
Lastly, my son went to a good private prep school, amassed very good stats, and didn’t apply to any of the schools you mentioned (no Ivys, no Caltech, no MIT). He didn’t see any of them as attractive based on what he wanted out of an engineering program. There’s WAY more to vetting programs than rankings and bumper sticker pride.