That first photo is rather confusing. Is the second more accurately showing an alcove over the fireplace?

As for hanging stockings, you don't have to have a mantle - there are these things, and they come in many styles:

True, thanks for the idea. I still don't love the look of the space. Would love to improve it

It's very true you could hang your stockings on the shelving near the fireplace..but it's just not the same and having a fireplace RIGHT THERE would accentuate that for me. My mantle uses stocking holders shown above. I would not want my TV recessed the way the alcove demands it be. A mantle is not difficult just to french cleat to above the firebox, if you did wall in the alcove. You don't need a mantle with side columns. The more modern look is just a substantial piece of wood across the top. Google "modern mantels". That way the tile job could be left for later. So where would you put a TV? Maybe you and your husband need to have that plan in place before moving forward? While you are at it will you wall in the alcove above your shelving unit? I've never seen an opening like that above what is suppose to be a 'built in'.

what is on the other side of the alcove?

We are stretched with our project budgets.

If you just moved in to this house, you will undoubtedly have unforeseeable expenses and repairs to make. Live here for six months to a year before figuring out what to do with this area. For this year, can you just use the types of stocking hanging devices others have posted pictures of or some other kind of temporary hooks? (I'm Jewish so don't quite get what the problem is with using the shelf as a mantle and hanging stockings from hooks for a year. Is it that you don't want them to lay flat against the tile and want them to project out more?)

Wondering if doors could be put on there to make it into storage?

I like it. Now I think I should hang my TV on the wall. Enclosing the alcove doesn't make the room feel Smaller, rt?

I like Carolb_fl_coastal's idea about doors, so here's something else you could consider if you square-off the top of the arch and install false doore above the ones opening below.

What @lisedv said makes sense. You can have something “deliberate” over the fireplace, have a place for your tv and utilize the large area to the right.

If you get someone in to close off the TV niche, have them close off the recess over the built-in too. It looks silly and you aren't going to reach all the way up there, and back, to display anything up there.

Do you have another place to watch TV? You could also add shelves in the current TV nook. The other ideas about the doors are also very good. I’ve seen this before and it ends up being the only place for the TV but that was because that is the only place they have and their space is really small. It looks as if you may have room elsewhere, but it is hard to see because we cannot see the whole room, nor do we know the dimensions. In any case best wishes in your new home and congratulations.