You say that a rooster with a no-crow collar can "breath just fine," but that is not always true. I've read quite a few stories of roosters that died because of no crow collars.They make no crow collars for roosters. You can buy them on Amazon among other places. They work by not allowing the rooster to fill with air before he crows so thus reducing noise produced. The rooster can still make all his other noises and breath fine just reduced crow volume. We have used these with success in the past. You can also YouTube it to see it work. It's just an option in case.
Apparently, there is a very fine line between "tight enough to make them crow quieter" and "so tight they cannot breathe." Some people and some roosters have good results, some others do not.
(I have no personal experience, so I'm going by what I've read, mostly on these forums. There are quite a few anecdotes on all sides of the matter-- works great for some, kills rooster for some, doesn't work but doesn't kill rooster for others.)