What should White have played in the initial position?

Given the course of the game, I originally thought that White’s rook should have just stayed on the back rank, and White should have just played Kg2 and pushed the h-pawn in the initial position. I felt that White’s initial Kf2 let Black develop his ferocious activity, so it was mistaken; however, (a) the centralized king that White has in the game is not worthless either, and (b) Black has some activity anyway:



However, it turns out that Black can still respond with Nb4, and after Nd3 White will still have to play Re4 if he wants to keep the e-pawn (and he does want to keep the e-pawn, right?). Given the success of the exchange sac in the notes, I also thought that maybe just pushing the h-pawn here would be good, with
54.h4 Nf3+
55.Kf2 Nxe1
56.Kxe1 and then if Black tries to play as in the game,
56...Nb4
57.h5 Nd3+
58.Ke2 Nxb2
59.h6 and the pawns are ready to carry the day; but obviously Black doesn’t have to be so cluelessly persistent about winning the b-pawn. It is ironic that these accidental-looking exchange sacs can both be so dangerous.