.44 appeal
A .44 mag revolver is largely a cartridge intended for hunting, though it had a strong following in the metallic silhouette game, but the ability to shoot .44 spl from them, or down load the magnum cartridges if one is a handloader, can tame the magnums and make them suitable for other tasks.
The question was posed regarding moose or bear, and I think the .44 mag would be marginal for moose or the big sometimes dangerous western bears. I've been around black bears a good bit in my younger days, and though there're a wild animal, and I don't fully trust them, they are not near the threat like their cousins, and a .44 mag would likely deal easily with the average blackie, and I suspect that such a bear would run under 300 lbs, though they certainly grow bigger.
The .44 is plenty of gun for whitetail and hogs and entirely suitable choice. Consider that one step down in the revolver world is the .41 mag, somewhat dormant, so the logical next choice is the .357, which will work for medium game certainly, but I would consider borderline in energy. Yes, lots of critters have fallen to the .357, but were I choosing a revolver hunting cartridge, (and I was, remember "your 3") it would be something bigger than .357. In the other direction, the .454 and .480 are bigger cartridges and begin to break into the realm of the speciality. Never shot either, but I suspect they are getting right up there in serious recoil. There are likely far more .44's in circulation and afield, than their bigger brothers.
The .44mag is no longer "the most powerful handgun in the world" and I'm sure many here can (and will) name others. The .44 is a handful, extremely noisy and with a lot of flash, but it is manageable and shot well by a lot of folks. Typically they are not shot a lot. One won't fire 200 full power rounds in succession at the range and claim they enjoyed it. But I think they can be managed by most enthusiasts. I would not hand one to bamawife, a younger kid, or any inexperienced shooter. But if you hunt with a handgun for medium game, want a poweful handgun in relatively portable package for protection from toothy critters, or play the silhouette game, they are the popular choice.