I think this is an interesting and useful discussion. I am a hunter first and prefer break-overs for that purpose.
I had a falling block Ruger No. 1 and it was a beautiful rifle. In general I think falling blocks are easier to make look good and I consider aesthetics an important part of any firearm, but some brak-overs also look good. The falling blocks are a lot easier to reload when shooting from bags on the range or like with most prairie dog shooting where speed doesn't count, but one does not want to have to move the gun a lot between shots. I think they are generally more accurate and cases generally last longer for one who reloads since the falling block actions are stiffer (but not necessarily stronger). Being stiffer and loading easier from a bench would make them better for a target or prairie dog shooter. At my gun club's rifle range firearms are supposed to be carried open with the barrel pointed up. This is OK with a falling block but very awkward with a break-over.
For hunting situations I think loading for a first or repeat shot is easier with a break-over. When the barrel is broken over, the chamber is very accessible, even with a scope on the gun, whereas with a falling block for me, with large hands, it's difficult to get my fingers under the scope to the chamber area. Although accuracy might not be as good as with a falling block, the right break-overs are more than good enough for hunting. I have trouble getting enough cheek support on rifles with scopes, especially high mounted scopes with large front lenses. With a break-over one can have a butt-stock a high cheek-piece for good cheek support. The high cheek-piece will not get in the way of the cleaning rod when cleaning the bore since the barrel can be removed for cleaning, also an advantage compared to a bolt action. Additionally, break-overs work well for having extra barrels in additional calibers for different purposes, which saves cost compared to having to buy a completely separate gun. In addition they work fine for barrels chambered for shotgun cartridges whereas a falling block action would have to be made very large for a shotgun cartridge. I don't think a single shot shotgun is very good for hunting game birds, except for turkeys, but it is fine for deer hunting with slugs, at least for the way I hunt them. Finally, when crossing a fence or small stream, or talking with others a beak-over can be broken over, with chamber loaded, and still be safe.
I use my single shot Thompson-Center TCR guns a lot for hunting and don't figure I have much need for a second shot. If I can't hit a standing animal with the first shot it would be unlikely to hit it running animal with a second shot. However, there is one big advantage of a bolt action for hunting compared to either a falling block or break-over single-shot. Where legal, it can be put in a gun rack in a vehicle with the magazine loaded but the bolt open and be fast to shoot but still safe.
