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That looks good

Why exactly can't white win if he converts his pawn into a knight or a bishop? :)

Because black can move Kb4 and then force exchange of pawns through Pa5 after which it's King vs King + Bishop / Knight with no pawns and from memory that's a draw.

That's completely correct! Congrats to you @miniature-tiger, the winner of this round! :)

It is on point though. Like your explanation.