"but using logic to convince people their religion is mistaken does not seem to work very well."

It worked for me. To be precise not "logic" but "epistemology" but you get the point. There are (few) believers who are open to reason and only haven't come around to thinking things through yet. I had to think about my religion for years and build a rigorous logical framework of christianity before I could identify the one faulty assumption on which all was resting.

Regarding the OP, I don't think arguing over minor points instead of the whole deal is futile. As I said, I needed to start thinking logically within my religious framework before I could break out of it. Of course I can't generalize from one data point, but still I am one example where this tactic worked.