Theoretically, Corte in the heart of Corsica gives you most options and it is a wonderfully placed and magical town in itself. Also, you'll have an ice-bath for free in the Tavignano river

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However, you'd probably appreciate some sea soaking on your rest days and surely you don't want to miss scenic coast rides which are a bit far from Corte. In this case the west coast is where you want to go. East coast is flat with long beaches but no cliffs.
I'd probably choose something right on the coast from Ajaccio up north. Sagone (good camping), Cargese (beautiful, great atmosphere but you have to downclimb to the sea) - both have 3 directions to begin the ride. From those to it's not that far to the Calanche rock formations and subsequent awesome descent to Porto!
But actually you can't make wrong choice, e.g. Calvi might be good as well and is a nice mid-size town with more infrastructure.
If you want to replicate Froome's performance on Col de l'Ospedale, you'll have to stay more down south.
Edit: just noticed you're looking for a "mountain base". If you mean base in the mountains, I think nothing can beat Corte. If you mean base for MTB riding, I don't know.