Some sheep stick their heads through mesh and others don't. I had a line of Soays which all had shortened horns from the habit, but the rest of the flock didn't break theirs.
Urgh, looking at that, yes it's a vet job, to get it cleaned up and the dangling bit removed. The danger with broken horns is infection, which has quite a direct track to the brain, so a broken horn can lead, via infection, to death. It's a shame as horned sheep are so attractive
From your description I didn't think it was so bad - a picture says a thousand words