Welp, she's working great now!
The problem turned out to be a simple one. Whoever was futzing around under the 6BG6 (probably before I was born), cut a cap out of it that I couldn't even see. Looks like someone back then was testing things and just gave up. That, and the width coil out of a parts set (thanks bgadow!) brought the set back to life!
Horizontal lin's a little off, but I'll play with that after a little usage.
And finally, another tough dog is done. Whew.