Didn't you already say the HZ meter and outlet were working? if so you are making power.
I've just skimmed over this whole thread briefly, but if sounds like you simply aren't getting K1 to close, correct?
If you have Hz reading, have a voltage reading and the outlet works then the only thing stopping power going to the lugs is the K1 contactor and its control circuit. Go back to post #15, did you remove the small wires and apply 24V to the x and y terminals? If the contactor closed then the issue is in the control circuit for the contactor, not K1 itself.
Granted you still need to determine what burned the wire, but at least you will know if K1 is working or not.