I don't know of any native ground covers that would grow well under an oak with that much shade. The only two that immediately come to mind would be horseherb (Calyptocarpus vialis) and dichondra (Dichondra carolinensis). Both of these fit your criteria, they're shade tolerate, native ground covers, that can take some foot traffic.

The most common plant I see under oaks is Asian jasmine, but this isn't something you would want to walk through.

One of the best things you can do is apply a good quality compost and shredded bark mulch under your tree. It will
combine with the oak leaves and gradually decompose to help amend the soil beneath in terms of drainage and access to the nutrients. It will have to be replaced from time to time, but that is still less expensive and work intensive than replacing plants that
will not survive in dense shade. Also, the mulch, up to 4 inches thick, will provide a non-muddy path and protect the tree roots from heat and cold as well as retain moisture in the soil.

One of my favorite (and most shade tolerant) wildflowers is Baby Blue Eyes (Nemophila menziesii). You could try planting these from seed along the edge of a shaded area if you want a subtle splash of color in the spring.

Best,
Brittany Taylor