Can I eliminate the expensive coax "multiple outlet" plug?

Yes you can do that, but you probably shouldn't. They do work, but you probably have an error in the cables.

1)make sure your amplifier is turned on.
2)try wiring direct from the output of the amp plate to the TV. Amazing how many times the cables are not even connected to the TV somewhere along the route.
3) check cable connections everywhere you can find them.