First off, if your dad knows a thing or two about med school admissions I’d value his opinion more than internet strangers.
Beyond that, I wouldn’t advise a recruit choose a school they wouldn’t otherwise want to attend just based on the sport (unless they are particularly passionate about continuing, or competing at a very high level; neither seems to describe how you feel about the sport).
I also wouldn’t pursue a sport for four years just for a med school boost, but that’s me.
Brown might or might not be a realistic option for you. It’s a tough admit for everyone. But in analyzing that I’d consider concrete differences beyond some vague notion of “prestige”.
Seems pretty straightforward to me: if you’d be happy at your state flagship and that’s a near certain admit, then the question is what advantage a near guaranteed admit as a recruit to these other schools offers. And are you willing to give up that advantage to play the Ivy/Stanford/etc lottery?
I think with that recruit option you need to “do the math” with the assumption that the coach leaves, you’re injured or don’t get to compete, or you just decide you’d prefer to spend your time doing other things. Also consider the possibility that your interests will change and you’ll drop the pre-med track.
As noted above, med school admissions for athletes can often mean taking a gap year, which isn’t unusual.