FWIW, the spotwelder I had in my shop used a standard contactor, of the kind that's commonly used for stuff like heanting and air conditioning to switch the transformer primary. The controls and logic were run on 24VAC supplied by something that looked like a doorbell transformer. The timing was all done with electromechanical plug-in delay relays connected in a cascade. They were calibrated in cycles, 60 being one second.
For a low-power setup like this, you could likely switch the primary directly with a foot pedal, but something industrial grade would be preferred, because there can be some big transients when the field collapses in the transformer.