1.gxf6! gxf6 2.f5+! gxf5 3.Kf4 and black is in zugzwang already. He will eventually lose all Kingside pawns while white keeps the h4 pawn and uses it as a distraction to buy time to munch the Queen side pawns. 1.exf6? gxf6 and now the attempt 2.f5+?? exf5+ 3.Kf4 fxg5+ wins for Black.
exf6 gxf6 then not f5 but simply Kd4 and Blacks best reply is Kf5, I thought Black would be in Zugzwang after that but it appears not. My Kd4 idea does work after gxf6 though, because then Kf5 exf6 Kxf6 Ke4 and that is the zugzwang we are after. Your f5+ line missed fxe5+ after Kf4