Almost certainly no. It would have to be really, really bad with other issues to make it necessary to put him down. I would certainly have a vet check him out (a goat vet, as mentioned) and make sure it's just his hooves that are the problem. Then, start slowly and work on getting his hooves into proper shape. Start by watching a couple of videos on YouTube and then take off as much as you think you can without causing him to bleed. Then, a week later, take off a little more. A week or two later - take off a little more until you have his feet corrected. I've heard some people say it took a couple of months to get a goats foot properly trimmed after severe neglect.
