Not saying that it is mental strength in Mayo's case par se, they are a couple of player short (although I do think the defeats do leave scars), but to say that the mental side of the game, and sport in general, has nothing to do with it is just plain wrong...
Look at the number of talented individuals in sports who never fulfilled their potential relative to their skill levels due to a lack of mental fortitude (John Daly comes to mind, there are many many others). On the other side, you have guys who absolutely maximize their potential through hard work and mental strength (Harrington, talented yes, but a mental warrior).
When it shines through is in the big situations at the end of big games. Cosgrave / COC / Charlie, didn't miss those frees/penos due to the lack of ability.
Baggio missing a peno in a WC final...Compare that to Pirlo doing a Panenka against England like it was a mess about at the end of training.
It definitely has something to do with it.