my state school Indiana University offers merit scholarships to around 20-24 of their top in-state applicants. around half of those are full tuition scholarships.
some of the "top" schools i interviewed at offer merit scholarships, typically to the 3.8/36 with impressive ECs kinda students.
the moral of the story with med school merit scholarships (perhaps different for URM, i would not know) is that you need to have some high numbers to go with a complete application.
there are different scholarships like for primary care and the military that are rather substantial--like full tuition plus stipend--but they do require matching years of service.
grant money does not quite flow as it does in undergrad admissions/scholarships. schools will not meet need like many top undergrad schools. there are some schools who are known for giving out a lot of grant money, but state med schools do not for the most part, nor do private schools dish it out like undergrad. my undergrad would meet need according to the FAFSA, which resulted in a 25K discount for me, but none of the med schools i visited were quite so generous.