I agree that you should paint the builtins.
I also would suggest you consider removing the fireplace mantel. It has a pretty flimsy stuck-on look and I think the fireplace would look better with something much simpler. More modern (assuming you’re ok with that).
I might also suggest you consider painting all the walls and trim the same white to de-emphasize the very busy trim all around the room.

Yes. Same white as the fireplace mantel and window trim.

Yes painting them would look amazing and lighten the room.

White. $1400 is very reasonable presuming he is doing all the correct steps. The prep will take as long as the painting.

From the pictures you posted, the built ins are not what is making the room dark. Your ceiling light is much too small for the space and casts the light towards the ceiling.

If your goal is to make the room brighter, replacing the ceiling light and adding more lamps would be helpful.

If your goal is to modernizen the space, the wall color and fireplace date the room more than the natural wood.

Yes to painting the built-ins white. But this will not totally solve your problem. The room is dark because you do not have enough lighting sources. The eyeball lights facing the fireplace are nice, but the only other lights you have are the table lamp and the “boob” light in the ceiling. I’d replace that with a chandelier that has multiple bulbs, and also add floor lamps. Also you could place a console table behind the sofa with 2 lamps on it.

I LOVE all these suggestions! Yes to All of them!!

I admit I know nothing about lightning. Currently I have 2 light fixtures in the ceiling. Can someone recommend What light fixture? Something in a photo you recommend? 1 or 2?

This is what I currently have

Lighting can be complicated. There are some great articles here on houzz. But for sure two overhead “boob lights” are not going to do the job well in a living room

https://www.houzz.com/magazine/10-ways-to-get-your-lighting-right-stsetivw-vs~10558866

https://www.houzz.com/magazine/a-beginners-guide-to-lighting-in-layers-stsetivw-vs~60963736

https://www.houzz.com/magazine/decorating-101-how-to-plan-your-homes-lighting-stsetivw-vs~12803740

Those should keep you busy for a while :-)

Thank you all so much!!

Agree. They should be white. Because they are in the casework/millwork category like the mantle. It's about color hierarchy and grouping all the case/millwork by the same color is one of those details that can have a huge impact on the overall atmosphere of the room. It will feel better not just look better because it makes more sense to have the case/millwork match in color/finish.