The soft-start thermistor being burnt, and/or the supply fuse or circuit breaker tripping is usually caused by excessive input current due to a shorted component in the inverter power section. This is often one or more of the MOSFET or IGBT power transistors.
In a half-bridge inverter the smoothing capacitors are in series across the rectified mains, and a shorted transistor will put full voltage across one capacitor, and if rated at only 200v it will blow.
When a power transistor shorts it can feed the high supply voltage into the gate driver circuits and damage them, and unless corrected this in turn will destroy a replacement transistor.
At the last posting the OP had replaced capacitors and a shorted IGBT, and his welder was powering up without blowing any further components, but would not arc. I would suggest you start by removing all the IGBTs and testing them for short-circuits.