What you say is true except he has no dangling wires.
Each one, the battery, the converter/charger, and the TV and a solar charger are energy sources and should not be dependant on the battery or any of the other components. Each one should be parallel and if hooked into the battery leads should provide energy to the system with or without each of the other components/energy sources, including the battery. IMHO. As far as the fuse panel receiving power from the battery, that much is true, but the fuse panel is actually connected to the battery leads as is the charger/converter, and the TV, AND solar if you got it.
The battery is just an additional power source and a energy storage device in the system.
If he is not getting power from the TV to the fuse panel without the battery, there is either a protection device installed somewhere or it is wired that way on purpose, or he has a broken connection.
It is probably a broken ground. DC does not like chassis ground, a chassis ground corrodes and creates resistance. I would find that chassis ground and run an additional ground via a copper wire (green insulated wire available at lowes) from that same chassis ground the grounding buss bar, located in the battery box. In fact any chassis ground that you find on your rig should be augmented by a direct copper wire back to that buss bar.
If his rig requires a battery to be installed before his TV can provide power to the rig then he is, without a doubt wired differently than I am. It could be wired that way on purpose.
Charge line from truck goes to the battery. The charge line from the convertor goes to the battery. The power wire from the fuse panel goes to the battery. Sensing a pattern yet? When all these wires are connected at the battery, you have a completed circuit. When disconnected, no circuit.