The mig welder must be a single phase unit then as capacitors are just there to smoothe the the power output a bit. The welder should work without those cap's but will spit a whole lot more, if there are several capacitors (there usually are) you could try taking the blown one out of circuit, I would suspect that if the contactor is closing when you pull the trigger on the torch that one or more diode's are blown. In my experience with mig welders (35 years) the most common fault is in the lead from the trigger switch down through the torch, the wires tend to break just short of the handle, depending on the torch configuration try shorting the contacts for the trigger switch this is much easier if the torch connection is of the euro type as the torch just unscrews as a unit a lot of the really cheap units the torch just goes into the case where the various leads diverge to their relevant connections, there is a US system where the torch is terminated in a series of brass rings of diminishing size to form a taper that then pushes into a socket and is held there with a clamp its so long since I have seen one of those that I cannot remember the name of the torch manufacturer. The open circuit voltage of most mig units is around 24 volts big units may go to 31 volts or so the voltage will droop a bit on load but not as much as you get from a mma welder (stick) as they are a so called constant voltage system the current is controlled by the wire feed speed which alters the arc length.