I think a board that comes inside would be better.
When you decide to grill and bring it out it will be clean.
When you bring it inside to clean it you dont have to bring it back out.

When grilling I do all my chopping of veggies inside then I go out with meat on a plate, toss it on the grill, head back in to return the plate and grab cutting board and aluminum foil. The meat comes off the grill into the foil on the cutting board and it heads back into the house to be cut up after it rests. I dont currently have a patio or deck so It is eaten inside or in our sunroom.