Yes, Rice is a very good example of a “T20” that would actually make a lot of sense for the OP!
Rice is also super into kids who are super into Rice. I think it is therefore hard to successfully “shotgun” Rice.
But if the OP shortened up that Reach list to just the colleges like Rice that really made sense, I think it would make it easier for the OP to write thoughtful and compelling applications for each of those.