will Goats eat it?.........they could descimate it at first then spray less later.
Goats won't eat mature stands of brush or blackberry vines. They will eat however re-growth. Also on a hill side, goats will promote erosion. You also have to have a real good hot wire fence so they don't go for greener pastures. Remember goats are not stupid. If they can go for the neighbor's nice expensive tasty landscaping, believe me they will in a heartbeat.
Steve, that is a puny little patch of blackberry vines. Man up and go in there with a Stihl fs250 or fs550 first with a whip head, cut all the little fluffy parts off and expose the canes.
Then come in with the blade attachment and mulch down the rest. Cut it fine enough to lay nice and flat. Make sure you use chaps that cover, well, the parts of you that you consider important. Those brush cutters throw chunks perfectly.
The resulting mulch will help to hold the hill. Come back when the vines have re-sprouted to about a foot in height (no more), and spray with round up if you must.
If they don't like Round-up, then tell them to tarp it. Still won't work, but they may feel good doing it.
Better to just come back year after year and re-mulch the area. Its a good way to keep your help busy in slack times.
Here is a picture of the brush blade which I use, it does a great job. Not only in mulch berry vines, but also in throwing chunks into my 'private' parts, windows, cars, cutting irrigation pipes, etc. etc.