Couldn't find my probe, so I did as you suggested. No spark at all, not even when held about 1mm away.
Here's a pic of the flyback....didn't realize while testing tubes yesterday that it seems like it's kind of baked. Not sure, but at the very least, it's overheated as the black coating melted off, some of which is lying around the surrounding area on the bottom.
So what would be the next step?
I supposed I ought to do as you recommned with the tube sockets, but after pulling the tubes last night I did work them back and forth in their sockets just in case.