Why does Lichess mark this move as "?!"? I was looking at this game on Lichess, where it says "Checkmate is now unavoidable. Best move was Kc4." As far as I can tell, checkmate was already unavoidable, including after Kc4. Namely, Black can move Ke3,Ke2,Rf3,Kf1,Kg2,g3 and then either gxh2 or Rxg3 according to White's next move, leading to a won end-game. Am I missing something?
There is no valid human reason. The difference between Kc4 (mate in 12), and Kc2 (mate in 10) is fairly irrelevant. It simply appended the "?!" based on the algorithm that 10 is slightly worse than 12. Either way, it is, indeed, a simple win from there.

By the way, my plan would be Rd7, then bring the king to h1/g1, and only then, bring the rook over via the f-file to take on h2. The longer you can keep the white king out, the less you have to calculate.

In general though, if the situation were different, and the material were more even, you might not want to box your king into the smaller area.

 [FEN "1B6/8/8/8/4k1p1/2Kr3p/7P/8 w - - 0 0"]

 1. Kc2 (1.Kc4)