Beautiful home! I love your sitting area in front of the fireplace.

It’s really personal preference. In a large open area like yours, I’d want one to anchor the table/dining area, provide a warmer feel, etc. But honestly, it looks perfectly fine without one, too.

Since you have the rug already, maybe put it in there for a few days and see what you think?

I agree that you do not need a rug under your table. It is anchored by the dog bed. Just kidding. It is framed/anchored by the windows.

I pondered this same issue, with a table centered on two patio doors, and decided not to use one. I don't think there is necessarily one correct answer, but I think your table looks lovely and lets the floors/surroundings serve as accents.