Mine do 'graze' but not as low as sheep do, (except just now when gras is very short). They do prefer branches and will gnaw bark off if they get the chance (managed to convince OH it was rabbits stood on the deep snow
).
IF possible I think it would be ideal to fence in half, make hay in one half then you can turn the goats onto that half when fresh grass has grown, giving them a boost just before winter. Even better would be into 4 and rotate grazing, but fencing costs money.
My front field is 4.5 and I wanted to divide it into 3 strips, but I could divide it into 4 quarters for very little more fencing, hmmm, glad this subject was brought up, made me think before I started wasting more money
.