Update: I actually took it to my guy, he ran it like a bat out of hell, and just like the car that won't make that noise for the mechanic, he couldn't get the thread to break. I brought it home, and now it doesn't like C&C either.
My assumption is whatever the issue is is between the thread and the needle, not between the needle and the fabric. I may have partially found the problem, but haven't got it solved yet. On the tension unit there's a guide to the left; the thread goes around the tension, through the check spring, then down slightly and back up under this finger. The finger has a groove worn into it where the thread rides. I'm thinking that groove is what's fraying the thread, and by the time it passes through the needle eye a few times, it just can't hang together. It's a mystery why it wouldn't do it for the fixit guy. I suspect I'm going to have to make a trip to Lowe's for super fine sandpaper, but I'm hoping that will solve the issue.