kraigwy is pretty close to spot on. Most cowboys were migrant workers or subsistence workers, and spending a month's wages on a revolver when your boss didn't allow it was not in the picture. Shotguns did a lot of the hunting work, and cap and ball firearms were still in use right up to the late 1800s. Small bore cartridges were the norm for deer hunting. At the end of the 19th Century and well into the 20th Century, the .32-20 was considered a fine deer cartridge, with low recoil and available in handy, easy to carry, inexpensive rifles that you could set by the back door to chase away the occasional varmint from the hen house or carry in the buggy to pot a grouse on your way to town. A Winchester 1873 cost a good bit of money, so not everyone could afford to have one just sitting around. Instead, there were a lot of older surplus weapons in use on ranches and farms, old Trapdoors, Spencers, and the like.
The 30-30 was an amazing cartridge when it was first introduced, 2,100 fps with a 170 gr bullet in a 6 lbs package was pretty cool, especially when you compared it to its competition, so it was no big surprise when it took the gun world by storm. Let's see, you could have a 1873 Winchester in 44-40 (1,400 fps with a 200 gr lead bullet in a 9 lbs package), or a 1873 Springfield in 45-70, or a Savage 44-1/2 single shot in a variety of small-bore cartridges, or a Winchester 1894 in 30-30 that would shoot to point of aim out to 150 yds and still had enough energy at 300 yds to down an elk. You could not buy a 1893 Mauser outside of the larger cities, and the lever action was king of the repeaters for many years, until after WW1 at least. After that, the rules changed, but the Winchester 1894 was and still is a great rifle.
BTW, most of the buffalo herds were slaughtered using black powder muzzleloaders and 45-70s, the military would sell ammo to civilians and if you wanted cheap ammo they were the place to go as long as you owned a 45-70 or a 45 Colt firearm, so that was what many commercial hunters used. If you owned a rifle in an exotic chambering, you had to special-order ammunition, which could take weeks to get, although mail order was generally considered pretty fast in rural America. Many of the cartridges we consider to be "buffalo gun" cartridges were in fact target rifle cartridges so they were very uncommon anywhere, and many were not even introduced until most of the buffalo had been killed off already.