The problem of this Weller is that sometimes the temperature adjustment works normally, but suddenly it just heats up to the maximum and have no adjustment anymore. The solder handle part have been changed from a functioning same model but the problem remains, so the problem must be in this controller board.

I would need a schematics and there are some available online by Google search, but none of those fits to this board. They might be another version or year model.
Schematics should include 16 pin 4053 IC seen in the pic, all of the ones I found did not include it.

 

That's 23 years old! Chances are its a dry joint on the board, or intermittent wiper on one of the pots. Reflow any joints that look iffy and give the pots a good squirt of contact cleaner, as well as the connector in case its an intermittent temperature sensor connection. If that doesn't help you'll need to reverse engineer the board - it looks relatively easy as only 5 IC (what are they?) and not much hardware for tracks to hide under...

 

In the first picture, is the lowest capacitor swollen? it maybe camera angle.
Good luck...........

 

I would measure the voltage across the large temperature preset and wiper , and when it's running check it again, also the voltage signal feeding the opamp from the iron sensor probably a thermocouple.

 

I found this manual only: Description; Control Unit; Technical Data; Starting - Weller WS 80 Betriebsanleitung [Seite 23] | ManualsLib