Thank you for the photos and the test results. I did not know that there was a Murex with two "Dual-diode Modules" - I thought they used either four individual stud diodes or the typical finned aluminium assembly with multiple pressed-in diodes. Those diode results look O.K.

Does the 13A plug fuse blow at all settings of the rotary voltage selector switch, even the lowest? When set to the highest position, some large MIG welders can trip an input circuit breaker due to the large inrush current on the main transformer. However, a 165 is not that large, and a 13A fuse is not as sensitive as a circuit breaker.

With one cable from the transformer secondary to the rectifier still disconnected, could you measure the AC voltage across the secondary at all settings. It should vary in roughly-equal steps from about 15 volts to 32 volts AC.

I have never seen a large welder capacitor fail short-circuit, but it is possible. There are six capacitors, typically 10,000uF 63 v, all connected in parallel via the aluminium bussbars. They are polarized and must not be reversed! Not all welders have capacitors across the DC output. They give a smoother arc, and also raise the output voltage. As Dragnet said, you could remove only the bottom right capacitor, reducing the total from 60,000 to 50,000 uF, and test the welder without that one.

To avoid explosions, large electrolytic capacitors have a rubber plug in the top. You could check if any have popped or are bulging or leaking.

To test a capacitor: a fancy digital meter may have a Capacitor test range, but it may not go as high as the 10,000uF components typically found in the capacitor banks of MIG welders.
Checking resistance on an old-fashioned analogue testmeter can indicate that a capacitor is not shorted and can hold a charge. The meter needle will initially swing over to the right, indicating low resistance, and then fall back to high resistance after the capacitor has charged using the voltage from the meter. Observe correct polarity - the black meter lead is Positive when measuring Ohms on an analogue meter.

Another method would be to use a 9 volt PP3 battery to charge the capacitor - observing correct polarity - and then measure voltage across the capacitor, which should start about 9 volts and then slowly fade as it discharges.