I created an interactive animation of a simplified version of the circuit using https://www.falstad.com/circuit/ website.  It shouldn't be treated as an accurate simulation, but it shows the general idea about how the welding pulse is triggered and how capacitor charging and discharging happens.

You can open the circuit in your browser by following this link: https://www.falstad.com/circuit/circuitjs.html?ctz=CQAgj...

Step-up converter module is ommited here.