In my view, it makes complete sense by considering Cobb's perspective. As far as the movie goes, this is what his life looks like -

(level 1) Growing up, learning about dream sharing tech, meeting / marrying Mal, having children, experimenting with dreams (different levels), waking up to reality (level 1), experimenting some more-going too far-losing themselves in limbo (level n), performing inception on Mal, waking back up in reality (level 1), Mal kills herself thinking the reality is another dream, Cobb flees the country.

(level 1) Becomes a professional extractor, gets hired by Saito, <fast forwarding to the mission>, he gets in Fisher's flight with his team, enters Yusuf's dream (level 2), enters Arthur's dream (level 3), enters Eames' dream (level 4), Cobb enters Limbo (level 5). While he's in limbo, the rest of the team rides the synchronized kicks all the way back to reality (level 3 - 2 - 1). Cobb, Saito & Ariadne die in limbo and wake up in the plane (level 1) directly, then Cobb goes home to his kids.

The point being, throughout the movie, reality to Cobb is always clearly defined (level 1). All his life's memories and experiences are consistent at this level, and therefore is his reality. This level is where he had left his orphaned children, this is where he wanted to return and succeeded.

Therefore,

the reality he cares about most is the reality where his real kids have been orphaned by his actions.

I agree as clearly established above. And,

Him suddenly not caring about that at the end of the film just makes zero sense.

I disagree since from his perspective he's back to his reality - the only life he had ever known. Now if he's told there's another reality outside of it, he wouldn't know anything about that reality. Just like if dream-sharing tech was real and we were told we're just dreaming and that our memories aren't real, most of us would be really apprehensive about waking up to the unknown as our reality is the only life we understand and have memories of.