When I played this endgame, I did not understand the subtleties involved. The computer analysis was telling me that 51. Ke3 was much stronger than the move I made, but at first I had no clue as to why it would have been stronger.
After diving deeper into this, it became clear that White needed to prevent 51... g5+ and 52.... Kg7 with tempo. Also, it is easier for White to win this game if he can roll his pawns with f4 and e5, because in some positions the Black king cannot let himself get exposed to Rf8+ and the Black rook is tied down to stopping the b7 pawn.
I don't know if these ideas are common sense or not. I was playing a speed game and I did not pick up on them.
In the analysis I don't show every possible variation nor do I explain every move, but I do show the above mentioned ideas.