The local company that I know of (Rent-a-Goat) is not willing to bring fewer than 450 goats (!!!!) to a job and I just don't think my neighbors would be cool with that.
Have you asked the company how long it would take their 450 goats to clear the amount of brush you have? If they do it fast enough (a few days or less), your neighbors might not mind.
I have several acres of poison oak, thorny brambles, and dense thicket.
I'm not really expecting the chickens to eat poison oak and brambles. If they can scratch it up enough to kill new growth, that would increase the amount of time before I have to hire an excavator.
I would expect chickens to prevent new growth in their favorite areas, and close to where they live, but not all over.
A chicken run with 10 square feet of space per chicken will often be bare of grass and soft plants within a few weeks, although some trees and bushes seem happy to live forever in those conditions. If you want to match that density over your several acres, that would be an enormous number of chickens!
Chickens are good at killing established plants. Has anyone used a movable fence for chicken-brush-clearing? I'm not sure I can even get a fence in, due to all the thick brush.
I once used chickens to kill some small areas of weeds. I used a chain-link dog kennel that was 10 feet on each side, with 4 or 5 hens in it. It could stay in one place for what felt like a long time, probably weeks, before moving it made any sense (if I wanted the plants dead.) That particular kennel had 4 sides that were attached at the corners, so I could take it apart and re-assemble around trees or bushes. I had some trouble with chickens flying out (it was 6 feet high), and they would go under the edge if it was on uneven ground (until I blocked the openings with rocks.) I shut them in a more secure coop every evening, because I knew knew that dog run was not safe enough at night.
You could try putting up a fence around a section of brush, add chickens, and see how it goes. Depending on the results you get, you might continue that way or change plans (expanded fence with more chickens, or goats, or people with power tools & machinery.)