Metal covered control surfaces enabled the Spit to turn faster and tighter, the development coming at around the time where countering the superior Fw-190 made this an increasingly important factor: You can read a mention of the vast difference metal ailerons conferred on the Spitfire in Johnny Johnson's fabulous autobiography ''Wing Leader'', when he recounts flying a mission over France with Douglas Bader. Bader, being the boss, of course had one of the first of the then-new Spits with metal ailerons. Johnson radios Bader and says something along the lines of: ''steady on Douglas, we haven't all got metal ailerons!'' when Bader puts on an extremely tight turn in combat and Johnson cannot stay with him.
Fabric covered control surfaces can balloon out in a vacuum and cause control stick force difficulties and under extreme circumstances they can even shed their surface, and of course they would also be more susceptible to damage from flames too, the doped surface being extremely flammable, so it is conceivable that strikes from incendiary rounds could render the control surfaces useless.
Well since the treatment was only to the ailerons it did not allow tighter
turns, per say, but it greatly improved roll rates at speed,
so pilots could theoretically initiate and complete turns much quicker leading to increased agility. The important thing about the upgrade and the debate that spawned it is that the ballooning phenomenon occurs at high indicated speed. Yes, from what I've heard the metal ailerons did improve the handling and feel in all regimes of flight but the most profound difference, as intended, was during high speed rolls such as during dogfighting.