MIG welder problem

Transformers almost never fail in this way. Usually if they are overloaded, one winding can burn open like a fuse, or it gets internally heated and burns off insulation which can cause inter-winding shorts--and more smell--and smoke. In other words it is likely OK

I would check any and all connections, including those on the ends of the transformer windings, and any other terminations.

I would suspect the diodes/ rectifier. NOTE!!! Regardless of "internet wisdom" you CAN NOT test large amperage diodes with an ohmeter. The ONLY reliable way to check large diodes, such as in welders, large current power supplies or even alternators, is to LOAD them. This might be difficult if you don't have "some junk". "some equipment". "Some knowledge

If you have a carbon pile battery load tester, you could use that. you could remove the diodes, hook them up battery--load tester--diode--battery, and adjust the thing to draw 40-50A or so. measure the voltage drop across the diode with a meter. You should not see less than 1V or not much over a volt drop if that

This might be a potted "bridge" rectifier, and gets more difficult but can be done