Indeed, the idea you used to win is useful, but your opponent played inaccurately. In many lines the win was hard to find. You might want to check that analysis by me (Elo 1800 in blitz on that site) :
Just take a look on the 42...Rb7 line and tell honestly if you would have found the win !
This is one of my first victories ever in a Rook + pawns endgame. After a long struggle, I reached the following position:
I think it is quite instructive how the principle of two weaknesses ( a passed pawn on the queenside and a better Kingside pawns, conveniently supported) applied in this position, forcing the black rook to take care of the passed pawn while my king, taking advantadge of his hanging pawns, marched up the board to swap roles with the rook(the king helped the passer to promote while my rook was looking at his kingside pawn).