Check the rectifier using the diode test function on your multimeter, it's just a beefy bridge rectifier, so can be tested as such.
The inductor is unlikely to be bad, besides maybe burnt connections, so check the actual mechanical joints for signs of corrosion or burning.
The capacitors are generally obviously popped when they fail due to the high currents, but you can just disconnect them for testing if you want, they are only there to smooth the output voltage for smoother welds.
If there are any burnt connections, I used to use electrical jointing compound when remaking connections (Nickel anti-seize grease can work too, don't use copper anti-seize on aluminium!). I'd file/sand the surface flat and clean and immediately smear with the jointing compound and make the joint nice and tight and I never had a joint burn a second time. This is especially important where aluminium conductors are present. Check ALL connections in the welding current path, as well as all other electrical connections around the place.
Also, check the main switch for voltage drop across the closed terminals for each power setting (set your multimeter to volts and test across the closed terminals, so you are looking for any voltage drop across the 'shorted' terminals), you should have a few 10's of millivolts max when pulling full current during welding as well as idle. If you see a couple volts or more, you have a faulty switch.
Do this check as described and let us know the results, a normal ohms test can show a good connection under no-load but the contacts can fail when pulling current.
It seems the signal to the SCR is what would be expected. The ripple is at 100Hz, which is expected from a bridge rectifier running at 50Hz and the 11V seems about the right level for a good turn on voltage.
It looks like wires 54 and 55 are the AC power to the control board, this would be rectified for the control circuitry on the board itself. This appears to be working ok as the circuit does function somewhat.
Wire 53 is probably what is detecting the presence of a struck arc so maybe trace that around the PCB and check for any busted components.