I would see if you could get a carpenter to change the shelving in the built-in.. Put TV in there and place sofa across from it for daily usage

At times when you want to watch TV with more people, you can always move the two chairs around to accommodate.

Place the TV on the bookshelf.

Place the sofa on the long wall as illustrated below.

I'd paint out the ceiling beam. It's distracting.

I suggest you modify the shelves in you built-in to accommodate the TV.

That would allow for a layout something like below with a sofa table behind the sofa to bring the sofa forward into the room.

unanimous!

Catty-corner the tv. Replace the mirror over the fireplace with artwork. The rug should extend to the chairs.

You can put a tv on the countertop of the built-in without doing anything to the shelves. Tvs are super thin. If necessary, add a piece of wood to fit the base. I think at least a 65" should easily fit. Then add a sectional with large ottoman.