I hate wood stoves. That said I have one in my family room. I am insisting that we build a concrete wall around the whole thing so all I have to see is the fire and door. If you can get it inside your fireplace do so it will look so much better. Forget the built-ins.

I was excited about the wood stove ("free" heat) ...until we had to find a spot for it.

When you say "forget the built-ins"...do you mean they should go? (are they ugly - I don't know anymore)

I can totally identify with you . We had the same issue in our home with too many focal points! It was confusing and difficult to solve. Our built-ins were on a different wall from my fireplace and so it was an even more difficult situation I think. Personally I like your built-ins. Its difficult to see in your photos but maybe they simply need to be updated with new hardware and paint. We decided to get a fireplace insert in the fireplace with a blower unit. Great heat! We mounted our 52" flat screen above it. I wasn't wild for the idea initially, but it did solve our problems and now that's its been in place a while I am very happy with it. Our room is much more comfortable with appropriate focus. Your room looks large enough to have a couple of different sitting areas. I would suggest putting a grouping of chairs near the bay window with a round table between them and use a sectional or regular sofa with chairs and sofa table near the fireplace and tv (if you also mounted it above the fireplace) A chimney company can easily check your flue to see it it will work with a wood stove or insert. The only think I wish we had done at this point that we did not is to put in a gas insert instead of wood. I've had wood fireplace, woodstoves, etc. all my life. I just sick of the mess. A romote control that turns it on would be so nice. . . . and tidy! I'm all about less housework!
I just glanced at your photo again and an idea came to mind. What about removing the smaller built in and using that space for an attractive (but faux) display of wood. I saw Candice Olson do that on her show and it was stunning. Food for thought.

You had mentioned that if you pushed it in where the fireplace is you would have to remove or change the built-ins. I would put it in the fireplace even if the built-ins need to be removed. They aren't ugly the wood stove is. I do agree free heat Yeah!

Love my Jutol wood stove and it heats most of my home. I guess it depends on where you live and if you have cold weather, I do. You need an expert to tell you if your fireplace is safe for a wood burning stove. There are also requirements on how far away the stove has to be from anything combustible materials.

The benefit to an insert vs woodstove is that you can use the space above the fireplace.
A woodstove (I fear) kicks off too much heat for that. I could be wrong. Our blower pushes the heat away from the wall and therefore the TV. Our hearth does not get hot and the tv is just fine. Believe me, I check it all the time.