Well the slight snare buzz, that's not even noticable when the band is playing. I never noticed it even when I record. And since I play acoustic jazz the sound I want is a nice high-pitched resonant sound. To have an open "un-gated" sound like that you're going to get a little noise, I guess, but if everything is tuned right it's just a part of the overall sound of the kit.
Very true, you will never be able to completly eliminate it if you want that specific sound. By the way, I envy you playing in a band like that, I don't even know any other jazz mucisians.