If the attacking king has already reached the sixth rank, the extra doubled pawn provides the spare tempo that a single pawn lacks, and the attacker wins.
Now the white king is on d6, already past the sixth rank, and the pawns on e5 and e4 are behind it. Against a single pawn Black would still be defending with Karstedt's checks from behind, and the attacker would often be unable to make progress because of a lack of spare moves. The extra pawn solves that problem.