As with other existing trees of some size and age doing things under oaks that predate development of the property can be detrimental. The stone patio might actually be less problematic for it than adding a bunch of planting, with associated bringing in of organic matter or soil, watering and fertilizing etc. As long as the patio is installed on a sand base, without a lot of cutting, filling or compaction, and space is left between the stones for water to get to the oak.

I garden on sandy loam under oaks and my favorite low and mowable groundcovers are the stonecrop type of sedum (I really love blue spruce sedum) and ajuga. just mulch mow the leaves over them in the fall or whenever you get to it and let the chopped leaves enrich the soil. They won't take a lot of foot traffic so wouldn't be ideal right around the table or in the path to that door. They stay really short, under two inches here and get weekly watering in the hot part of the summer here north of Fort Worth.

Cedar sedges can be inserted as plugs. One can rake up the leaves and not damage the plants. I also have a lot of phlox pilosa, cedar sage, penstemon tenuis, salvia lyrata, Heartleaf scuttelaria, I have lots of things growing under my live oaks.

I love your wall treatment on your shed. Nice surprise with the galvanized. My house is Galvalume and and scarlet pink hardy panel.

Pots! I will plant stuff in pots (caladiums or other shade loving plants) and place them under the tree. You can place them in a ring around the tree and when the plants fill in, it almost looks as if the plants are planted in the ground.

A stone patio for the table, with a few large potted plants would look really good. Horseherb grows under trees and everywhere else at my house. For flowers, Mexican petunia (invasive) and yellow columbine do well under my oaks.