Why did Luke care that Han and Leia would die?

At that time, Luke was a kid who started out lonely, and then lost the only family he knew AND his mentor AND his best friend AND he left everyone he'd ever known behind, shortly before "TESB".

Humans are social animals all the way down, and it's necessary and healthy for a human to form close familial, social, or romantic bonds with other humans (don't get me started on the Jedi), and Han and Leia were the people he had the closest bond with of anyone that we know of. They were his friends, his co-workers and allies, his "family of friends", possibly he still wished Leia was more than a friend, they may have acted as mentors as well as friends because they were older and wiser (respectively), etc. Of course he'd care if they lived or died, losing them would have been absolutely devastating.

He's the hero. Even Dirty Harry would've gone back for them.

They were his friends, why would he NOT care?

The same reason he wanted to turn Vader back to the light side, instead of just eliminating him.

He knew Vader was trying to draw him out by hurting his friends, and that’s GOT to have given him a lot of guilt. Yoda and Obi-wan were wicked cold in expecting Luke to literally turn his back on his friends. I disagree that Luke leaving his teachings was some sort of pupil failure, because his fighting Vader gave Leia, Chewy and Lando an opportunity to escape, and Luke even waited to go back to Yoda until Han was rescued.

they owed him money

They were his friends.

Because even if he knew it was a calculated risk, he's not an emotionless detached unresponsive sociopath.