I agree with your strategy, as long as you’re happy with your in-state flagship (and no reason that you shouldn’t be). However, another thing to keep in mind is that you may achieve a higher GPA (and hence have better acceptances to med school) at your in-state flagship than at a highly competitive ivy or equivalent.
There’s really NOTHING on which your family could better spend the 200K more it’ll cost for private college over your in-state flagship? In your situation, the research opportunities will be there at the public, the perfect GPA might be more easily obtained. You will probably get into just as good a med school from your in-state flagship as from a prestigious ivy.