As long as the items being broken down netted LESS materials than what was initially put in, it would help (though certainly no return of mats would help more). The problem that I see with making it a class specific thing is that there might not be as much of an impact as it could have have if anyone could desynth. That said, I REALLY like the idea of a salvager as a class, especially if they had repair abilities like you mentioned. The question is what ELSE could they do that is worth while? Aside form being able to repair stuff, what do they actually craft? I'm not sure I can think of anything, to be honest, especially since items don't actually break in this game (Which is good, because OMFG, breaking fishing poles in 11 got expensive!).
Hmmm... If there were more clock-work or tech type stuff in the game, you could call them Tinkerer instead, and have them specialize in stuff like that (though it might infringe on GLD and ALC a little). Perhaps parts like pullies for compound bows. Clocks as furniture for moghouses (once we get them). Bits and bridles for chocobos? Flint lock / matchlock mechanisms for muskets. Locks for gauntlets and swords? Obviously anything for articulated tools; maybe parts for a loom for weaver off-hand alternative, or mill/press for cul. Just a couple things that come to mind.