I want to be clear. I am not suggesting a correlation between LSAT score and law school exam performance. This has been disproven as a motivating factor in school selection repeatedly.
A hamilton or rubenstein recipient is not just a set of numbers. Only a couple dozen are handed out each year from a pool of many hundreds of 174+/3.8+ candidates and the selection criteria include additional indicia of high performance and ability beyond how well you padded your college GPA with political science seminars or how many powerscore classes your parents could afford. These won't be enough (although there's no set recipe for that 'something more', it could just be growing up in mississippi for example).
One odd responsibility that comes with the territory of involvement in campus life is that I've seen hundreds of resumes of incoming and current 1L students (who list their merit scholarships rather conspicuously) and they come from great schools, have additional degrees, substantive work experience, consistent leadership, publications, fellowships, ect. And they "do very well" in a more general sense at the school because they can impress and perform academically and otherwise in ways that make substantive differences both to their law school experience and future employment. At a school where you'll get a great job nearly regardless of your grades, career advancement takes different shapes. I'm not talking about raw striverism so much as drawing on prior experiences, expertise, creativity, maturity, or whatever else.
Does all of this combined help "get better grades" too? I would say probably yes, but in an environment where everyone's bright there's no hard and fast rules.