If you can plant the rhizomes without disturbing the oak roots, you could plant a row of cemetery irises along the sidewalk. They are EXTREMELY drought-resistant and will gradually fill in spaces over time.

Then, you could also plant some purslane between them and the trunk of the tree. It's another very drought-resistant succulent (that's also edible and highly-nutritious).
http://serenityinthegarden.blogspot.com/2010/10/magnificent-purslane-edible-landscaping.html

Here is a link that might be useful: Iris albicans

Might try putting a layer of mulch over the bare areas and watering it in. Keep the layer light (say 1/2" thick) and you may see your grass fill back in. You will probably see some roots surface near the trunk of the tree as the tree ages and the roots grow in diameter no matter what you do.

We have two larger live oaks about 30 yo and the grass cover has waned and waxed over the years. The grass under dripline of our oak trees is St. Augustine where it is protected somewhat from sun. The rest of the yard is either Bermuda or Buffalo.

yes.. grow the grass.. and your bare soil issue goes away ...

adding soil above tree roots is not really recommended at much any depth ... though it sounds you are aware of that ...

in fact.. most likely ..... that is the soil that is washing down.. so why put it back???

ken

ps: if you dont.. or cant water enough for the most invasive weed in the world.. grass ... then i really doubt you are going to find some other crop that will thrive under that tree ... drought tolerant plants ... not unlike your lawn .. are not drought tolerant.. until well established.. and nothing will get well established in a desert .. so you will have to water ... so then its just a simple decision as to whether you want grass.. or some other plant ....