The problem is that you're not being consistent in your handling of unfalsifiable theories. A lot of what's been brought to the table are Russell's Teapot-type problems and other gods, but I think I can find one that's a bit more directly comparable. I'll present a theory that's entirely unfalsifiable, and has a fair amount of evidence supporting it. This theory is that your friends, family, and everyone you know are government agents sent to trick you for some unclear reason. It's a theory that would touch every aspect of your life, unlike a Russell's Teapot. There's no way to falsify this theory, yet I assume you're assigning it a negligible probability, likely .0001 or even less. To remain consistent with your position on religion, you must either accept that there's a significant chance you're trapped in some kind of evil simulation run by shadowy G-Men, or accept that the impossibility of counterevidence isn't actually a good argument in favor of something. (Which still wouldn't mean that you'd have to turn atheist - as you've mentioned, there is some evidence for religion, even if the rest of us think it's really terrible evidence.)
First of all, in an intellectual debate, you don't go around telling someone that they're cornered. That ought to raise all sorts of red flags as to your logic, but in fact I'm perfectly happy to accept both of those propositions.
I would quite agree that there's a chance worth considering that I'm the center of a government conspiracy. (It's got a name.) I don't have any idea how that chance actually ranks in my mind, and any figure I did give would be a Potemkin (a complete guess). But it's entirely possible.
the impossibility of counterevidence isn'
If it's worth saying, but not worth its own post (even in Discussion), then it goes here.