Hi Nicole, welcome to permies.
I don't yet have
experience with this plant or much anything desert related, but I looked into brittlebush and had a few ideas.
I think it would be an easier
sell with your neighbors if there was no extra labor at all, or maybe some labor with potential to be paid for it.
Have you tried burning the stems? Another name I found was incienso, because a resin from the plant is used to make incense. Perhaps the stems can be burnt without alteration. If not, maybe they can be ground up and mixed with powdered charcoal, hydrated and formed as cones and allowed to dry, or some other type of incense.
There are also uses such as glue from the resin, "disposable toothbrushes," selling seeds/cuttings, and some
medicinal qualities.
Perhaps one or a combination of these qualities will help entice your neighbors to take the job for themselves, or allow someone to farm that part of their property, with the major benefit of them needing less water for their yard.
This way you may deal with them at different stages, at different times of the year, to spread the tasks out.
I'm not sure of a low tech way to prune the dry stems. I would maybe carefully stepping sideways on the branches to break them at a nice height if I wasn't trying to harvest them for more than mulch. A cleaner look could probably be accomplished with a hedge trimmer, or maybe a pair of large pruning shears.