So your radio that you have has a bad pilot light socket.
Wires crumbling off and shorting.
This can be fixed with a minimum of trouble, as finding an exact replacement nowadays is quite difficult given the different number of pilot light mountings and sockets out there.
If you look in the socket, typically you will find an insulating disk with a metal contact in the middle of it. There is a dimple in the side of the socket to keep this from flying out.
Carefully "squish" the dimple enough you can get the base contact out from the top.
There is a spring inside to provide tension for the pilot light. It is under the base and the wire(s) go through it.
These springs can break or lose tension over the decades. Pretty much any spring of the right diameter and construction can be used, provided it can provide enough tension.
They can also be quite rusty and that can snap the spring wire too.
Take some steel wool on a swab and gently clean out the inside of the socket, being sure no wool is left in it.
The base contact can be unsoldered and a replacement wire soldered to it, and to the shell of the pilot light socket. Some bases use 2 (bipin) contacts too.
The base contact can also be crimped to the wire in some types of sockets. Just cut the wire off leaving enough to solder to, and solder new wire on, then a short length of heatshrink tubing.
Feed the wires back through the socket. carefully dimple the sidewall again, and you should be just fine. It should look just like the original.
What? The base contact is bad? well, you can often use the contact out of an old car tail light socket or side marker, or the contact out of an inline fuse holder. You gotta get creative sometimes, you know?
This should solve almost all pilot lamp problems.
If it's used in a parallel circuit and the original lamp is a # 47, use a [You must be registered and logged in to see this link.]
bulb. They draw less current, and that can prolong the transformer life due to less load. They also make LED replacements, but those are quite expensive.
In a circuit like the AA5 radios, the pilot light is an important part of the filament circuit, being run off the center tap of the rectifier filament.(35W4 for example). If the pilot lamp is bad, it can shorten the life of the rectifier tube.
Some radios used a colored lacquer to tint the bulb. General Cement (GC Electronics) made a specific kit to dye the bulbs, wan back when. I have been told fingernail polish may work, but have not tried it myself.