I spent yesterday afternoon clearing out pesky native vines on 1/3 acre in Florida with 52" Ratchet Rake on the bucket.
The Ratchet Rake is not a ROOT RAKE but it does pretty well pulling out dormant vines by the roots, especially after a rain.
The Ratchet Rake is not a LANDSCAPE RAKE but it does pretty well raking/piling vines after they are pulled.
At $299 the Ratchet Rake is hard to beat.
After I piled the vines, I picked up the three piles one-at-a-time with 2,000 pound capacity Payne's aluminum Debris Forks. To load debris forks I pushed each pile against a tree used as a backstop. Picked up 20% dirt along with 80% vegetation but I did not have to get off the tractor. (Remaining on tractor is a safety issue. Always a small chance a hydraulic line will let go under max load.)
Each pile was 800+ pounds as I could barely lift them
I had 600 pound Howse disc harrow as ballast in the 3-Pt. lift or the front end would have gone down instead of the load up. Ballast behind rear axle reduced stress on the front axle.
Because of "danglers" I transported loads in reverse to the burn pile.
One or two passes with 48" King Kutter Rotary Mower (bush hog) in a few weeks should complete demise of the vines and scrub oak volunteers.
An enjoyable, productive day.
If you are using your chainsaw a lot consider purchase of 28V battery powered, reciprocating Milwaukee Sawzall. For anything up to 2" the sawzall is faster, blades are cheap and SAWZALL IS A LOT SAFER. If you acquire a PTO chipper you will use the Sawzall to trim branches before chipping; instant on/off is very convenient.
I know nothing about GOATS but I like the idea in principle. Can your rent goats?
I know you said you did not want to spray but if you have something that is large enough to use a chainsaw, or even a large pruners, pour a little Tordon around the edge of the stump to kill the roots. You do not have to coat the stump, just the outside edge. It is cheap and even though it is a chemical, local application like this does not contaminate things and you will not have to re-cut. You need to apply the Tordon very soon after cutting - like 5 minutes or so. you can buy Tordon at local farm store in a quart sized squeeze bottle with pop up squirt cap so it is easy to apply.
I've done very similar clearing of overgrown pastures and used a variety of implements. DR brush mower is great for smaller areas but gives you quite a workout manhandling it through rough terrain or more than an acre or so. Bush hog is easy but not so useful in wooded areas or when trying to be selective. Ratchet rake works nicely in small areas and can be very selective but doesn't work efficiently in large areas. Flail mowers can be offset which helps get close to trees and fences. AgriSupply sells the very popular Caroni TM1900 for much less than what JD or others charge. For Escape sized bushes, nothing beats a grapple and strong loader.
I use all of the tools mentioned in addition to chainsaw and trimmer. Each has pros and cons. For a large area I usually mount flail and grapple. For smaller areas DR mower or ratchet rake.
We do like the idea of goats and my wife and I talked about goats before posting the question here. But it's not fenced, we have construction going on, we're raising 29 chicks, adding a puppy, and taking on another critter right now might be system overload.
I was out there shoving on it just now with the front-end loader, so I can see where a ratchet rake or Piranha would be a great tool. After looking at it again, there may be too much of it around trees and rocks and uneven terrain to use a DR Brush Mower, especially the tow-behind kind. I'm back to thinking brush cutter and poison. A mix of measures might be the ticket. I'd like to have one of those brush mowers but hate to drop $3K right now. May end up with one, though, as a tow-behind mower would be nice for heavy stuff. Is the DR considered a flail mower?
i would look at a used brush hog $600-800 range 4' probably for your 1026 - 5' likely a bit big.
The reason for used is you have rocks and stuff so getting one used you wont feel so bad clipping a few rocks.
Also you can raise it up and back over brush cutting it down to size.
The DR is not a flail mower, but the flail mower is decent around rocks, but wont knock down heavier brush like a bush hog.
Here's one used John Deere 1050 4x4 tractor loader back hoe in NJ..
I have a Gravely Super 8 with 30" brush cutter that I might be will to part with but 300 miles may be a bit of a hike.. and a lot of work to run in your conditions - it wouldnt get the big stuff but would work on smaller vines and underbrush.
The model of handheld I use is a shindaiwa T2510 with a 12" saw blade attachment. Its about as heavy as a weed wacker, but has a ton of torque and runs at a lower RPM so it's pretty quiet. It is excellent with vines because you can swing it around. The handheld is better for working close to trees rocks and drop offs than something that is wheel powered. I think the total cost is about $250 or $300, but there are definitely cheaper models out there.
Keep the idea of goats in the back of your mind then. They will do a better job than any mechanical or chemical device. The seed from upwind will always be landing on your property and goats will be there to eat the new growth. Given enough time and goats, they will turn any woods into a park and keep it that way.
I had places I was afraid to back a bush hog into, never knowing if there were rocks or a 47 Plymouth bumper hidden in there. The goats cleared it out and kept it that way.