It doesn't come down to the best stick irrespective of aircraft?
No.
That's an extremely simplifying point of view as wrong as say "it's only the plane". In fact it's both. Or do you think I'm posing the same threat to the enemy when flying a I-16 as when I'm flying a Tempest?
No, I'm not. The Tempests speed, firepower, flight time, accelaration, climb do make a world of a difference, and that's basically why I currently have a K/D of 23 in Tempest vs 4 in I-16. In a I-16 I'm much more limited, almost every fighter is faster (and thus can disengage more easy), many of those faster fighters do even turn better than me and so on. I can't even catch most bombers in AH in that fighter!
In the end, it's the combination of pilot skill and plane performance. A better performing plane is like a better tool. If we were to express it mathematically, it's (pilot skill x plane performance) vs (pilot skill x plane performance). And the more similar the pilot skill of the opponents is, the more important the plane performance envelope will become (and vice versa)