Why capture with the King and not the Bishop?

With the queens gone, the king is better in the center for the endgame.
Castling is no longer needed. Kxd2 connects the rooks just like O-O does.
At d2 the bishop is worse than at e3.
At d2 the king is better than at e1 or g1.

The most important thing is that the evaluation for the 2 moves are tiny, less than half a pawn. Trying to understand these small differences is not productive. As the analysis gets deeper, the 2 moves can easily change places at any time. The wise thing is to assume that the 2 moves are practically equal in value, and move on with the analysis.

The computer makes assumptions about what black will play down the line - and that's when it becomes apparent that the K is better off taking.
But in order for it to be valuable, black has to play those lines - which you don't know in this situation. The importance could be 20 moves from this position.

The computer makes assumptions about what black will play down the line - and that's when it becomes apparent that the K is better off taking.
But in order for it to be valuable, black has to play those lines - which you don't know in this situation. The importance could be 20 moves from this position.
I have a hard time figuring out what you mean by all that. It sounds nonsense, but maybe you just phrased it badly.

With the queens gone, the king is better in the center for the endgame.
Castling is no longer needed. Kxd2 connects the rooks just like O-O does.
At d2 the bishop is worse than at e3.
At d2 the king is better than at e1 or g1.
I agree with this. Kxd2 is just a common sense move and I think would be the first instinct of anyone sufficiently highly rated (say around 1900 on ELO scale).