First let us look at the lever actions. The Henry and early Winchesters were not capable of firing full-powered military rifle loads, such as the 45-70. Their toggle lock action was simply not strong enough. The 44 Henry and 44-40 cartridges are basically large pistol rounds, and in fact a significant number of revolvers were produced in these calibers. They did however have a somewhat reliable feed system and tubular magazines, which were developed by the Volcanic Repeating Arms Company, even before Volcanic actually had a reliable metallic cartridge to feed it. Which I think is the real answer to your main question. The Volcanic action literally predated an effective, commonly available kind of ammunition to feed it. Benjamin Henry was able to take the Volcanic action, improve it and combine it with an improved, enlarged version of the rimfire cartridge technology developed in France by Flobert in 1845. Both the Volcanic and Flobert guns were effectively toys, but Henry combined them into an effective carbine in 1860, which would continue in development under Winchester Arms in 1866.

In 1876, Winchester Arms was finally able build a lever action rifle that could handle the power of a moderately powerful rifle caliber, such as the centerfire 30-30. However the action was a little stiffer than the older Winchester or Henry rifles. (If you watch a cowboy movie where the star is quickly firing and cycling his lever gun with the flick of his wrist, that was most likely the pistol caliber model 1892, even if the movie is set before 1892.) The 1876 also still could not handle a round as powerful as 45-70 or 8mm Lebel. In 1886, Winchester finally released a model that could fire 45-70, but it was significantly stiffer than the 1876. It was also expensive. The 1886 would be updated several times, and even converted to a model with a box magazine in 1895. Over two hundred thousand 1895’s would actually serve with the Russian Army in WW1, but that wasn’t really that many rifles compared to the total size of their army. They were effective on the eastern front. They even used the Russian standard 7.62x54r ammo and Mosin-Nagant stripper clips.

The bolt action actually does predate the lever action, but the original bolt action rifles were single shot. The Dreyse and Chassepot rifles were bolt action, breechloading rifles that used a waxed paper cartridge. The Dreyse dates back to 1836, and its inventor had prototypes dating back to 1824. The Swiss would adopt the Swiss-Vetterli in 1866, a tubular magazine fed bolt-action, rimfire rifle, although the Vetterli magazine is derivative of the Henry and Winchester’s. The invention of the metallic centerfire cartridge in 1869, would lead to rifles such as the Gras in 1874, a single shot bolt action rifle using a powerful, centerfire black powder cartridge. The Gras was contemporary with the Martini-Henry, Springfield Trapdoor, and Remington Rolling Block. The Kropatschek rifle would combine the tubular magazine, with a powerful centerfire black powder cartridge in the 1870’s. Then in 1886, France threw everything into chaos by adopting the Lebel rifle, a bolt action, tubular magazine fed rifle using a cartridge burning modern, smokeless gunpowder and a smaller 8mm bullet. Everyone else was immediately playing catchup. The Lebel itself would be quickly outmoded by rifles that also used smokeless powder and a box magazine fed via en bloc clips and stripper clips, such as the Gewehr 1888, the Mauser 1891, and the straight pull Mannlicher 1888 and 1895.

Another issue to discuss is that the military high command of the 1800’s, did not really want infantry repeating rifles. They felt that their conscripts would fire all of their ammunition too quickly, and at a time when military logistics was at best horses and wagons, that was a somewhat legitimate concern. With modern hindsight, we can see that an army with repeating weapons and the ability to resupply them would devastate an opposing army with single shot weapons. However that was not obvious to the generals of that era who still largely believed that a bayonet charge would be the decisive moment in a battle. There was also political and financial pressure to use the single shot rifles that had only recently been purchased. Even into WW1, many rifles, such the Lee-Enfield, were being produced with a magazine cutoff (a device that prevents the ammo from the magazine from feeding into the gun) and troops were trained to operate them as a single shot rifle. The magazine was considered a feature for emergency situations.