I guess ONE downside I might point out for using thumbhole stocks for hunting is how they carry when slung around the body, rather than just over one shoulder. In general, a "standard" stock has a pretty smooth bottom-line (belly side), so they ride very comfortably when they're slung cross body (over your head like a seatbelt). Thumbhole stocks usually have a very abrupt pistol grip, so they don't ride flat against your back.
There are a few ways around this problem though. The easiest answer is to carry them over one shoulder, rather than cross body. I have mounted the swivel stud in the base of the pistol grip, which makes the rifle ride slightly lower on the body, so the belly of the forend and magwell lays nicely against your back, with the pistol grip and buttstock actually extending beside you. Another option is to use a sling that carries the rifle with its SIDE against your back. Another option i've used is to tie the bottom of the sling through the thumbhole, again, lowering the carry position.
My favorite option is a "safari type" sling, so I can carry the rifle low-ready with a little extra support, but still sling it to cross fences, open gates, etc.