Sorry you and your goat had to deal with this. My Dad had cattle and the one and only time I've seen a teat cut that badly was from one cow who was a deft barbed wire fence crawler (guess the ditch grass looked greener to her?) and hers looked like that. Any adjoining barbed wire fences for cattle she could have gone through? However it happened, glad she will be okay!
I am not sure I would dry her off. I realize that that side will get very full and that might be a problem, but if you dry her off she may form the scar tissue and there will no way to keep it open. If she is an important milker to you for next year I would dry to milk her as long as possible and hopefully keep that open. If the vet is going to insert a teat cannula I would watch closely for mastitis, another reason to keep her in milk, it will keep that teat cleaned out.
On the flip side, if you dry her up and she loses that side don't despair. I have a doe that raised trips this year and twins last year with 1/2 udder. She had mastitis a few years ago and I didn't keep milking her and scar tissue formed and while she makes plenty of milk on that side nothing comes out and it eventually dries up but the other side gives plenty.