Vjeko,
standard failure mode for old Magnastat Wellers is contact oxidation/damage on magnetic switch.
Contacts develop resistance, and there is no full power applied to heater. It will still regulate at idle but have no power..
If you carefully disassemble handle part (mind the wires) you will see contact part of Magnastat switch. Switch can be replaced by itself. It is very expensive, you can buy complete well working modern chinese soldering stations for the price. You might try measuring contacts, and if they have resistance, try to remove switch, carefully open the switch cover and try to see if you can clean them. In my experience not very likely to succeed, but if you have nothing to lose, you might as well try....