Given I am currently using 75 ohm coax to feed my random wire, I
believe I could use a 300 ohm to 75 ohm (4:1) TV twin lead to coax
converter and get away with it.

Please correct me if I am wrong, but I think I would attach the random
wire to one terminal of the 300 ohm input. The other would go directly
to earth ground, or via a gas arc tube to ground.

On the other side, I would attach my coax cable.

To give the coax feeder a ground at the antenna end, I would to open
up the converter and solder in a ground wire on the coax sheild
terminal. This would kill off common mode currents.

This would give my antenna feeder system a matching transformer plus
give the antenna a direct path to ground.

Theoretically, I am I missing anything?

mike