I do care a lot, i never skip cut scenes and i never really got why people do that. I wouldnt feel like i got the full experience from the game that way
I like some games more than i like most movies, The last of us, Metal gear games (NOT Phantom Pain), witcher 2 would have been shit to me if the story hadnt reeled me in, Witcher 3, which i thought was amazing i actually cared about finding Ciri because she was Geralt's daughter and i believed it.
Or old games like Shenmue 2 i really was looking forward fighting Lan di and avenging my father and was sad when i realized you dont get to do that in that game.
Ive noticed people in here prefer silent protagonists like divinity where you choose your name and everything, but i love games like the witcher 3 where the main character has a name, dialogue and a personality, that might be the one thing i disliked about Persona 5, game was awesome but i would have really liked if he talked and had a personality of his own, i mean the name you choose is irrelevant nobody ever mentions it at all.
I always thought people who dont care about the story prefer shooters and stuff, not saying thats you TS.