I got in late but let me try some comments
1) that fried wire is likely due to someone changing the primary voltage from 240 to 120 (I'm guessing since you said the lights dim, that it's running off a 120V line). My guess is it has nothing to do with the issue you're observing.
2) Cleaning the socket pins is a good idea, as is tightening up the plate connections that you said that you've done.
Now that you've gotten as far as you have (reseating the 3-500, cleaning the pins) put the cover back on and see if the filaments light up when you have the amp ON but without any drive?
There's not much use going beyond that point.
Be very careful with this (and any) amp. Keep in mind that plate voltage can stay in the filter caps for quite some time after you've shut the power off , depending on the condition of the bleeder resistors and the internal resistance of the capacitors.
Let us know what you find....
I agree with you Syfr 100 %. The connection in question looks like a re-connection from 220 v to 120 v and the melting was caused by a heavy duty soldering gun. Also notice rosin specks on the chassis, that is a good indication of a soldering for the voltage change.
Syfr is also very very correct about the high voltage capacitors holding voltage for a long time after the amp is shut down... handle with great care. Sure there are bleeder resistors but still the voltage is slow to discharge.
With something like this I would dis-able the high voltage totally because that doesn't seem to be the problem. Dis-able-ing the High Voltage makes it much safer for you to poke around in the unit. But you will still have to use great care as to where or what you poke.
If I were faced with this problem I would apply a very low amount of RF drive to the unit and put the Amp in the transmit mode and use an RF probe to see if the RF is getting to the input of the tube (cathode). If RF is getting to the tube the next thing to check would be the connection of the tube grid to ground, if that connection is OK then.... errrrr... the tube might have a problem with the Grid... no Grid, no amplify... That's never a good thing, in the past I have vaporized the Grid element in tubes... Yes it can be done and much easier than you might think...
Check the relays, mechanical things tend to be a very weak point in any device. Look at the contacts of the relays they should be shiny but if they are black then they need cleaning. If they are silver then they can turn black just by sitting around... many years ago I used mercury to clean up relay contacts but mercury is something you really shouldn't play with. Sometimes relays just get tired (the springs and alignment) and don't do what they should.
I have a gut feeling the RF drive just isn't getting to the tube so check the schematic and look at the RF input flow, all circuits and connections from the RF input to the cathode of the tube. If RF is getting to the tube and the connection from the tube's grid to ground are ok... then the tubes grid might be the problem.
Anyway.... have a Merry Christmas and a happy new year!
later...