I'm not a fan of cutscenes. As the joke recently reiterated in Borderlands 3 -- "You never die unless its in a cutscene."
The stories in Classic were mostly emergent. Read the scraps of paper and books; talk to NPCs; listen to the dungeon and raid boss emotes. You experienced everything in the context of gameplay.
The Butcher in Diablo was awesome, because he was unexpected. You're grinding away monsters, then you open a door and a room that looks like a murder scene has a unique mob in it...he yells "FRESH MEAT" then charges right at you. This was 100x more effective than any cutscene or exposition in Diablo 3, as they tried to ingrain MMO mechanics into boss fights (including rage timers and boss pull countdowns).
Classic's story was simple, and directly experienced. This is why players feel connected; its THEIR story, not the one the developers are trying to force down your throat.