I have both types.
I like felt the best. It gives better traction, unquestionably. And I've never felt the need for studs with felt. (Rubber soles without studs aren't just inferior, they're practically worthless. At least the ones I've tried.)
But I do think that the environment comes first, and I don't want to be the one who spreads an invasives infestation to a new stream.
So I have more than one pair of felt soled boots/hip waders, and unless I'm fishing the same water, I wait at least a week before using the same pair twice. Unfortunately, I don't get to fish as much as many of you.
I'll switch to nothing but cleated rubber soles if I have to.
Ultimately, I think the answer is to have several pairs of boots, because even rubber soled boots can spread tiny diatoms like didymo. They just clean easier because the soles dry faster, that's all.
Given the amounts of money I spend on other fly fishing gear, I don't think it's excessive to spend money for a few extra pairs of boots.
I've read about some boots that have removable felt soles, which would be perhaps the best solution of all, if the manufacturer would sell the boots with a few extra soles included. Felt soles can be completely soaked in salt water (NOT chlorine bleach!), they dry faster, they can even be put in the freezer (if you're a hermit, or insane and don't care who knows it) to kill any traces of infestation.
Ultimately, controlling the spread of invasives comes down to everyone being very conscientious about cleaning all of their gear between streams and between uses. Rubber-soled boots aren't going to solve the problem all by themselves.