I rewired the probe and put the whole gadget back together. Powered it up and the eye tube didn't glow. Took the gadget apart again and found that of three tubes in parallel, only the first one was working--no filament voltage to the other two. The original solder connections were "stick the wire through the hole, then glue them together with solder." I redid a lot of the connections from scratch, re-melted and added solder to most of the others, and so far it seems to work well.
One more thing. I know Electronic M wants to see this. It seems that the electrons were confused about which was to go when they got into the probe. So, to ensure optimal function, I added a reproduction of the original Heathkit decal. (And cleaned the aluminum, renewed the plastic, and polished the screwheads--all for a tool!)
Thanks for all your help. Now I can try to locate the noise in my TV, which is why I started this whole thing.