Chinese elm for sure. Beautiful. I like Fantex ash and Australian bottle tree too

It's a little unusual, but how about a Jujube? Not good for heavy shade, but a very cool looking tree.
RSI Growers might have a nice tree left in stock.

Elm and ash aren't desert trees, so I wouldn't plant those unless you plan on watering your gravel landscaping.

My sister has an ironwood right next to a lawn area and it's doing fine. They have nice flowers in the spring, but unfortunately some seeds, thorns too.

Are you familiar with desert museum tree? nice. It's a hybrid.

I had a similar question... what desert-adapted shade tree works in a lawn setting?

There aren't too many options, but the article below has some tips.

I finally settled on a Sonoran catclaw (acacia occidentalis) which is doing great so far. Except it's only 3 feet tall - I had to start it from seed, because no nursery carries them. But, *eventually*, it should turn into a nice shade tree.

Here is a link that might be useful: Desert Trees in Turf Areas

Desert trees won't survive in a lawn. They'd grow tall and fast and either die of rot or be constantly broken by the wind in their low dense pithy wood (from overwatering).

You want xeriscape do it all at once. Piecemeal won't work. Now you could plant desert trees and totally cease watering the lawn as a non-HOA approved compromise. The eventual and I do mean eventual (slow growing desert trees) shade would further kill off any surviving non-native grass.