You need to have a very frank talk with your parents about money, and run the net price calculators on your schools (other than the in-state publics). Can your family easily afford 85K/yr for 4 yrs for college? Would the extra 200K that you would save by going to U Va as opposed to that private T20 that you might get into, not matter one whit to you or your family? You have to realize that you might not win any merit scholarships, that T20 schools mostly don’t award merit money, only fin aid, so you really need to know whether you would qualify for fin aid, and how much your parents have set aside for college already for you.
I’d say that U Va is at the top of your list, you have a very good chance of being accepted there, and it’s most likely to be your best option. If your family would qualify for enough fin aid to make privates affordable (and that’s a big if - cost of living in Northern Va is high, so I would imagine that your family’s household income, in order to live decently there, is probably high enough that you would get little to no fin aid), then I think that you could probably get into many T50-T20 schools. With a carefully crafted application strategy, you MIGHT get into a T20.
Engineering is tough. So many people slog their way through it undergrad, and then wind up not using it at all, wind up going into other fields. Do you really want it? You’re already not planning to go into it, even before you start. If you’re absolutely sure that business is your future, why not major in it undergrad, instead of planning for the MBA? You sound like someone who might not need the MBA at all, might go on to start your own business, and the undergrad degree would give you the background in pretty much everything that the MBA would, so that you’d understand accounting, marketing, etc. enough for running your own business. I’d only plan on an engineering major if you want to go into a business that is engineering-based, so that you’d be able to have some understanding of the technical aspects of the business that you’d be running, or if mechanical engineering is your obsession, and it is what would make you happiest as an undergrad.
Honestly, in your position, unless your parents are so well off financially that the cost of college doesn’t matter at all to them, or so poorly off that you’d qualify for enough fin aid to make a generous private cheaper than UVa, I would only apply to the in-state public safety that you’d be very happy to attend, in the unlikely event that you didn’t get into UVa, and apply to U Va early decision. You’ve clearly worked very hard in school to achieve so much, could certainly benefit from a T20 school. But if you and your family are one of the many who are in that huge “in-between” area, financially, such that the cost of private college is burdensome or out of reach without big fin aid, but you won’t qualify for big fin aid, then you are very lucky to be in-state for such a fine school as U Va.
If you really love mechanical engineering, and don’t want to apply ED to U Va, consider adding RPI to your list. It’s very good for mech E, as engineering goes it is 2nd tier, gives generous merit money that often brings the cost of attendance close to that of your in-state flagship, has been known to then even match your COA at your in-state flagship including the merit you might have been awarded at your in-state flagship, upon appeal. But it’s less of a full experience socially than U Va would be, since it tends to attract a male, engineering-obsessed student body (70:30).