Please show us the whole wall in the daytime.

Right now your TV looks like it is floating in front of shelving. You need to make it a home in the Space. Instead of creating a hole in the center, make things off center and add visual interest.

Get doors all the way across the bottom.

I am not comprehending how a TV worked in the old hole, but no matter. Maybe take the built in out? You seem to have little to display and no books. I can't tell if your shelving is adjustable or removable. If so as suggested a few shelves gone and the TV can be set into it which would be preferable to what you are currently doing.

The only thing you can do with this without changing the shelving at all would be to try to find a small piece of furniture with a couple of drawers that would fit into the space.

With minimal change you could take two of the shelves that are behind the TV and move them into the space aligned with the current shelving. Remove the existing trim on the shelf above the doors and replace it with one piece going all the way across with the new shelves in place. If you could get a set of doors for the front center it would be better than open shelving.

No matter what you do the new TV is always going to look awkward floating in front of the shelves.

These shelves are stationary - someone anchored them to the walls, but that does not make them permanent. They really are not that hard to remove and you can do it yourself at zero cost. You may be able to use one or both sides in another room - maybe the babies room for some extra storage - a place for toys and books.
You could use both on opposite sides of one wall and place a single shelf between the two and create a desk for future homework time.

You could add a vertical line to each side of the tv, just beyond where it ends, by adding wood that has the same width as the bookcase sides (looks like 2 by 8's or so). Paint the boards, then cut to fit, slide in place on each shelf. You'll have a small opening on each shelf that you can leave open or cover with painted foamcore.

Try making a panel the size of that space below tv. You could attach the panel with magents & or heavy duty white velcro tape, so you can attach & or remove to access. Easy fix. ;)

You could do this with minimal woodworking skills:

Excuse my lousy paint job, please.

If you could give us a closeup of the shelves and measurement of the depth of the shelves and height of each shelf, we could talk you through extending the shelf across the open area. I'm thinking a 1 inch thick board, a thin board for the facing, 4 L brackets to attach on each side, and matching paint and some glue to attach the facing.