Way back when I went to a top law school, I was from Podunk U ( a state flagship but not a particularly prestigious one). Most of my classmates were from Brown, Amherst, UCLA, etc. Which anecdotally says 2 things, yes, they take people from a wide swath of undergrads, and also yes, it makes a difference the name of the school.
I started working in NY and then moved back to a top firm in a regional city. All of the young attorneys I worked with or who clerked for us were either in the top 2% or so of the 2 local law schools, or went somewhere more prestigious. I had been working there a couple of months before human resources realized no one ever asked for my transcript, so they asked me for a copy to put in my file. That would not have happened if I had gone to a different law school.