I guess I wasn't very clear in what I was trying to say. I think that if the OTA works on all TVs, and the cable signal, which flows along the same wires should also work.
Here is the diagram of how the booster connections look.
You say the cable input (the middle wire in the diagram) to the booster works when directly connected to the BR TV. What I am suggesting you try is to disconnect the cable input line from outside and the TV signal output line (the wire on the right in the diagram) from the booster and connect them together using a barrel connector. This totally bypasses the booster and will tell if it is the problem or not.