Okay, I guess I found what's been causing this. I followed a seemingly unrelated error listed in error.log, which claimed that Widget Logic, one of my oldest and most trusted plugins which I've been running on both sites for years, seemed to produce an unexpected ';' somewhere. I deactivated Widget Logic, and gone was the crawling error!
I've deactivated each plugin before, it was the first thing I tested for, but I guess I must have skipped that one because it's so old and trusted and is running on both sites.
I haven't found the erroneous ';' yet (or the gap in my conditioning that would cause a ';' to be erroneous), but if I keep not finding it, I can just reopen this thread in the Widget Logic support section.
Thanks for all the help, Mika! I appreciate you taking the time!