I'd go with a coffee table and no area rug over the carpet.

It's a personal choice. Either option would work.

Agree on the ottoman. We have one as well. Would not do an area rug over carpet, but if you change to would floors put one in that has your favorite colors. Almost anything would work with your couch. If you want additional color in the room consider changing out some of the couch pillows for colorful ones or by slipcovers for them. I would also hang a picture or tapestry on the wall behind the couch.

It seems that there needs to be a table somewhere for resting beverages. Some lamps would be good as well, maybe one floor and one table. Is there hardwood under the carpet? Some ottomans come with a reversible hard surface.

You have the beginnings of a very nice room. I'd base my decision on an ottoman vs a coffee table on the size of the room. It's hard to tell from your photo, but it looks like an ottoman might dominate the space between your sofa and TV. Personally, I like area rugs over carpet. They add instant impact and interest to a room. I've done it many times and have seen it done successfully by designers on Houzz. https://www.houzz.com/photos/2013-southern-living-custom-builder-showcase-home-rustic-family-room-phvw-vp~5772511 (The photo is one example, you can find others by doing a search "area rug over carpet".) Your room needs a pattern somewhere, and because the room is very neutral, a little color would pep it up. The area rug can do both. You can see in the photo how all the accessories in the room pull colors from the rug. The painting, the pillows and all the little things they have scattered around. This room would be boring without those accents. Good luck and have fun.

Thank you everyone, for your opinions. I have some direction now.