So the good news is many colleges are on what is called an exploratory model where you just start taking classes and don’t actually declare a major until after a year or two. Where they have major subdivisions, you’d probably want to start in something they might call Arts and Sciences or similar, which is where most of the popular majors for premeds would be, and also a variety of other options that would seem potentially suitable for you.
It looks to me like your big constraint is cost, as $40K is below the full pay amount for most OOS publics and privates. That’s fine but unless you really have a strong preference for leaving the state, you should probably apply to a good selection of your in-state California options. You have a great system in your state, and many of those colleges would seem to fit your requirements. And even with merit, I am not sure you will get comparable value at many OOS publics.
Miami (Ohio) though is an interesting idea, since they do have merit. You should check if they would get you on budget, but it has a different feel from most publics and could be worth considering. Pitt is also an interesting school with merit, although I am not sure it will get on budget for you. Another I like to mention is Minnesota, and their NPC will actually tell you your cost with merit.
In terms of privates, Case does have merit and seems like a good fit for you. Rochester would be another I would recommend, and they actually have a pretty cool and flexible curriculum structure.
If you want a more likely option, you might check out where the University of Denver would get you with merit.
With colleges without merit like Yale and Columbia, you need to check their NPCs to see if they would be on budget. These are fine for exploration, but I would suggest you see if Columbia’s core curriculum is really to your tastes–it is exploratory in nature but less flexible about exactly what you explore. If you want to explore more freely, you might check out Brown which has an open curriculum.
Another good Reachy college for exploratory kids is WashU, which makes it easy to combine all sorts of different interests. Lots of premeds at WashU too, which is not necessarily a plus per se, but the advising is very good for kids who go that route. Again, though, you need to check the NPC at WashU, as their merit is very rare.