Did I Miss A Draw?

ROOK endings are notoriously drawish, both in theory and in practice, but what do you think of the following position?

Black to make his 36th move in Thomas Brown (2068) - Spanton (1954), Hastings Masters Round Four

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Firstly, we can dismiss 36...Rxc6?? 37.Rh5+ as completely hopeless for Black.
In the game I played 36...Rd1+? 37.Kc2 Rd6 38.c7 Rc6+ 38.Kd3 1-0
But, as TB suggested afterwards, I should have tried 36...f3! The point is that 37.gxf3 Rxc6 brings about an example of the infamously drawn f&h-pawn ending.*
TB was not sure if the actual position on the board would be drawn, but the Nalimov endgame tablebase confirms the position is indeed drawn.
White could try 37.Rh5+ Kf4 38.Rh4+ Kg5 39.Rg4+ Kh5 40.gxf3, but again the f&h-pawn ending is drawn after 40...Rxc6.
So, yes, I did miss a draw (assuming I would have played it accurately).
*If you do not know about this, consult almost any general endgame book.