I haven't ever seen that happen. The only time I can remember seeing this happen is when someone didn't have their power transformer screws on tightly, then didn't get 0 ohms on the resistance check at the end of the manual for the end lugs of each 6 lug strip, so they added in wires to connect them all. That sent that particular kit on the fritz but it was OK once it was returned to normal.
When you first turn the kit on, the 6080 doesn't conduct because the cathode is cold and needs to be heated up by the filament. Generally your symptoms could be explained by a gassy 6080, but that ought to go away with use and shouldn't be an issue when you have a replacement tube. Since you have two tubes doing the same thing, that does suggest an issue with the amp itself, as damaging a tube to the point of making it gassy is going to present many other obvious issues.
Maybe a fresh shot of the wiring or two will reveal something we have missed thus far.