Replace the socket

I re-read your post. Are you sure the lamp driver board is good and sending a ground signal to the bulb?

If you soldering iron is plugged in, its possible the tip is at ground, which causes the light to illuminate. The braid is the source power, and the wire is the switched ground.

The take away is Classic Bally/Sterns lights are fucking horrible! They just are. Sockets are cheap as shit, replacing them doesn't always work, sometimes wiggling them fixes everything

As Billc above says, note which connection does what. If you mean soldering iron on the power braid makes it work = socket.
If on the ground then issue coming from the board. You can use a wire & jumper any other ground on a socket that works and see if it makes this socket work = ground.
Classic Stern...probably both...sometimes Just throw the Yopp in now.

I like my games to look modern and factory at the same time. I want LEDs but am going for the factory look. For that reason, I don't use Yoppsicles. The sockets The Pinball Resource sells are the best replacements available if you want to go sockets. I just replaced every socket in my Star Gazer. Every single LED worked first try. I didn't need to fuss with anything. It costs a little more to do it this way, but I feel the end result is better.
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Not crazy, just standard socket stuff. There's a space/wafer that goes betwixt the socket and the body that keeps them isolated from each other. These go bad over time.

I added some photos.

Either way works well, personal preference.
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Look, even with Yopps there's a lamp out

I actually like how the Yoppsicles light up a frosted lens. The clear ones look a little funny in my opinion.

All Warm White Yopps; And if you go Sockets my advise is to go all warm white.

I replaced the problematic socket strips in my Freefall with yoppsicles. I could’ve used regular sockets with LEDs since I’m using an Alltek lamp driver, but being close to the edge on one side and close to the ball walker mech on the other, it was cleaner to go with the yoppsicles.
I’ve since bought 555 boards to replace them so I can reuse the yoppsicles for another project where a LED lamp driver board is not an option.
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I agree, with one addition. Under translucent blue inserts (like the spinner and right stand up on Freefall), I use Comet ice blue to really make them pop without being "too" blue.

No I didn’t make a restoration thread. I was late getting started on it for TPF 2020 and it was A LOT more work than I anticipated. Could’ve made one after the show was canceled, I suppose.