Let me start by saying I have no idea what is inside your Hobart stick welder, but I am guessing that 20 25 years old, there are no complicated "electronics" inside, just "electrics" in other words, transformer, rectifier, chokes, and capacitors. and of that group, the most likely to fail are rectifier diodes, and capacitors. diodes most likely fail by shorting out, and caps fail mostly by going open from drying out. of course they can short out also, but the failure mode you describe sounds to me like an open cap. You can test them with an analog Volt Ohm Meter (VOM) by taking a known good cap with the same capacitance rating and placing the leads on each terminal and noting the "kick" of the meter (goes to low resistance them falls back to high) in a certain time period. and then testing the suspect cap and noting its kick if any. reverse leads and test again. reverse it again and test again. Testing in this manner is subjective, you must get a "feel" for the action of the suspect cap as compared to the know good cap. if it does not kick, but shows a constant resistance, it is bad. if is shows shorted, bad if it shows open both directions it is bad. The alternative is to get a real capacitance tester, but they can cost some money. But by testing with the VOM and just a few minutes to educate yourself as to what to expect, you can do a fair job of deciding if the cap is bad or not. Chokes should show as a low resistance. Chokes with the current ability I would expect in a welder would show pretty much as a dead short to a VOM. maybe 1 or 2 Ohms. Also while in there I would check out any switching contacts, for burned look, and contact resistance (should be a dead short). So there you go, I would tear into it myself, and have a look, you have nothing to lose. Oh, you may be able to get parts at either Mouser or Digi-Key, good luck
James K0UA