None really affected me, but it's precisely because of death quotes and to see where students end up if I don't recruit them that I wish this game had more route-exclusive characters.
I got a switch and Fire Emblem only this year, and my first playthrough was Silver Snow(on accident). Because I recruited no one outside of the Church, I ended up being able to field a (mostly) Church Based squad, which I thougth was cool so I wouldn't get tired of the main Black Eagles lineup for my Crimson Flower playthrough in the future. It was interesting to see where Ashe and Lorenz ended up, although I don't really remember any other characters showing up, specially because as you fight the Adrestian Empire while having all the Adrestian empire characters outside of Hubert and Edelgard, there wasn't a lot of room to see former students.
In my second run, Verdant Winds, I recruited Ferdinand and Linhardt because I now knew I needed them for paralogues. Seeing Bernadetta and killing her was interesting; Caspar I couldn't see a single fight or death quote because that map made me desperatly rush the Death Knight in Maddening. The most dramatic lines were when I had to kill the Blue Lions in Gronder Field 2.0. And I really liked Dimitri calling their names as I killed them. Except for some reason he didn't call out Sylvain's name (and Dimitri died before Mercedes and Dudue, so I don't know if he sticks to the "Name..." trend).
Had the game incorporated more students in their houses and into the war, with more route-exclusive characters to supplement their absence, I think they could have written more meaningful deaths for the students that don't join you.