I’m not sure where you go matters for getting into med school. You mentioned top 20.
Vanderbilt is a top university hospital. Look where their residents went.
Neurological Surgery - Miss State, Rutgers, Ok State, Quinnipiac, Pitt, Penn State, Florida A&M, Miami
Radiation - UTK, Santa Clara, College of Saint Benedict, Sewanee, Ole Miss, UMASS
Yes, some come from top 20s to like Rice and Vandy.
How about find the best place to excel for four years so the experience is great - which translates into strong grades and hopefully a top MCAT.
Personally I don’t see where, at this top hospital, that school name is making a difference.
If you want top 20, it’s up to you. But it doesn’t appear the magic pool of entry has much to do with school name or rank.