Seems the Chihuahua desert has a full suite of hardy nitrogen fixers, to name a few
Catclaw Acacia
Honey Mesquite
Trailing Indigo Bush
Coulter's Lupine
Tepary Bean
Partridge Pea / Sensitive Pea (Chamaecrista fasciculata / Chamaecrista nictitans)
Crotalaria
Check out Seed Source or google them, there are producers for tepary, coulters and other legumes etc in bulk.
https://seedsource.com/hidden-categories/ecoregions/Chihuahuan_Desert_Trans-Pecos
I think along with reseeding, the methods Brad Lancaster has been presenting for rangelands and drainage would work for that region, these use
low profile geologic structures to slow and stop erosion, sometimes lift the flood plane back to the bank, stop headcuts, spread sheet flow into shrub tickets... It has an accumulative effect on the whole landscape. A good resource of how to's on Zeedyk structures or
low profile geologic structures IE (Zuni bowls, rock runs, media luna, check dams, leaky weirs) is Quivira Coalition -
https://quiviracoalition.org/technical-guides/. They have great guides for how to implement these on landscape.