John,
Good job on the measurements and understanding how this all works now, you may be an RV electrician yet.....
Somewhere between the power source and the transfer switch you are not connected on one of the hot lead sides. Good news is that the transfer switch looks to be fine. When you say you had to take apart the power receptacle, I do not know exactly what the function of that receptacle serves. If it is the one that the main power cord gets connected to coach wiring in, then by all means that could be the problem. Short leads and wire nuts are notorious for having issues if they are not tight or the wire is not fully inserted. They should be tight enough to twist the wires together and as a precaution then the wire and wire nut taped together all together with electrical tape so they cannot loosen up. If you are talking about a standard 120V outlet then that is not your problem.
If you can gain access to where the power cord is coupled to coach wiring in a junction box, that is where I would go next. If you get 120V on each leg of the main power cord then your problem is beyond that point. If not it is the power source, adapters, the cord, or the plug. Keep us posted, hopefully you are almost there.
Later Ed