The K1 has nothing to do with the burnt wire.
The burnt wire could have been there a long time, unless you took a look in there before.
Did you remove the bonding strap before powering your house?
The Troubleshooting diagram can not cover every possible problem. Thats what the TM is for. Instead of tapping the K1, take it out. Follow the instructions on how to bench test it. In fact, you do not even have to take it out. You can bench test it in the set.
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Remove wires 150B and 152D.
Attach 24 VDC to the terminals X&Y.
The contactor should close. you can test it with a DMM.
Also, if your DMM has roach clips" (Alligator clips) attach them to the X&Y terminals and start the set up. Use the S5 to close the K1. Hold it up. If you do not get 24 VDC at X&Y, then the problem is not in the K1 and you need to look someplace else. If you get 24 VDC at X&Y, then the K1 is the problem.
TB4 is a Terminal board. That's a place to connect lots of different wires for different circuits.
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I will guess that the wire that burnt was 100G. Coming from PP Ground. If you did not remove the bonding strap on the set. this burnt wire could be the reason why.