Our Ultra D.C. Burnisher starts instantly when the cord is plugged in, even before pulling the safety trigger levers. Can you suggest a reason for this?

Use your volt meter to test the battery voltage. It should be at least 12.4v . I this is good, follow the voltage through the ignition switch and the relay if it has one, on to the starter. If you have power on to the starter, but still no crank, the starter is bad.
Hope this helps, Robert

What exactly is the question here?

Is this a corded electric Scrubber or a battery one?

Plug it in,pull trigger

The connectors are that thick the cable is thin.

www.clarkeus.com

Ohm out the electric cord. Sound like an intermediate open. Most oftem broken close to the end(s)

https://www.kenteuroclean.com/service/manualspartslists.aspx

www.mercuryfloormachines.com/burnishers.aspx

Air traped inside carburator or lines.