I think a rug would look nice in this room ... it would anchor the table and give more definition in such an open space.

I like your ideas of adding red. I did some mock-ups of how red would look, as well as jute rugs. But I don’t know anything about the living room not shown, so can only go by these pictures.

(In mock-ups, the colors tend to look a bit washed out, and I used smaller rugs than you’d need. Doing big rugs in mock-ups tends to skew the patterns.)

https://www.rugsusa.com/rugsusa/rugs/rugs-usa-olden-tribal-medallion/Red/200RZIN05A-P.html

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https://www.rugsusa.com/rugsusa/rugs/surya-srp-1001/Dark-Red/158SRP1001-P.html

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A rug with more muted colors ....

https://www.rugsusa.com/rugsusa/rugs/rugs-usa-cartouche-medallion-tassel/Multi/200SVIN13A-P.html

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This jute in real life has a darker gray edge than the mock-up shows ...

https://www.rugsusa.com/rugsusa/rugs/rugs-usa-stallion/Natural/200NCNT02A-P.html

A plain jute ...

Thank you Jinx for your response and suggestions. I love all of your choices but especially the first 2. I have looked at so many rugs I was second guessing everything. I needed a second objective opinion. Thank you.