whether it’s the wall or the cabinet, you need a hole! otherwise you have to run the cable outside the cabinet and leave a door slightly ajar FOREVER. it would drive me mad….

You can drill through the floor and run it through the crawl space. Or you can tack the coax cable neatly down the side of the cabinets and along the baseboard around the doorway into the other room….. That’s how it’s done.
If you are having trouble getting the wire through a hole in the floor etc…, you push a clothes hanger wire through the hole, tape the coax cable to it, and then guide the coax cable through the hole.
If you drilled a hole clean through to the outside of the house and tied into cable there, that’s how you get the cable through the hole.

There are wireless devices which allow you to receive something you broadcast from another room, but they only broadcast one signal at a time. You might also need an RF modulator with this device. I recommend you hire a handyman to run some coax cable from the TV to a cable junction in your house.