Has Milly got Anglo Nubian in her - just those ears look like one.
My present goats are not keen on bark of any kind, although they eat the leaves of all sorts of trees. Just make sure they do not eat acorns off the oak trees, and any evergreen or privet hedge.
I feed my goats either goat mix, or sheep mix. In winter they have soaked sugar beet added, plus some of the ponies alfa a or chaff. They like most veg including peelings, bread, etc. Some of them like polo mints as a treat. I have to watch them with the hens feed, as mixed corn is one of their favourites to steal.
Hay is very important, and they do tend to eat a lot, especially in winter. I find they browse in the field, prefer trees and bushes to grass if they can get it.