An important preface to this discussion is who was the gun buying public, and what was their intended goals with said particular firearm. By and large the discussions around long arms prior to the 1980's in civilian hands was focused on "game shooting", or what today we'd call hunting. Martial use of firearms wasn't a very common topic of discussion, and when it was discussed it was limited largely to pistols. Martial use of long guns was the domain of the military and law enforcement, and if an everyday citizen was looking for a martial long arm this is where they'd be influenced.

There's also two major parts of the gun buying market, Continental Europe/the British Isles, and North America/Africa. In Europe and the British Isles, the majority of the land was owned by large estates, and the rights to game was given to the person who's land the game resided on. This meant that game shooting was the pastime of the affluent, since you would have to be invited by the estate owner or purchase the shooting rights to that property. In the Americas and Africa, there was the general practice of "free shooting", meaning in the eyes of the law the game belonged to no-one until it was shot and then legally would be the property of the person who shot it. This meant that to the European, the firearm was a luxury item, but elsewhere in America and Africa the firearm was a tool. That's not to say farmers in Europe didn't have firearms, or the American affluent didn't aquire luxurious firearms, rather it influenced the markets that American and European firearms manufacturers targeted when marketing their wares. Americans and those in colonial Africa needed strong, dependable arms for subsistence hunting and possibly having to deal with angry, displaced natives, whereas the European buyer was more willing to buy less proven designs, possibly less reliable designs in order to have the best shooting, most modern designs available. Now that we've defined the markets that manufacturers were targeting, the two common themes that will continue to come up are cost & speed.

When Casimir Lefaucheux patented his double lug break action shotgun in the 1840s to use his pinfire shotgun cartridge, he basically defined the format of the modern side-by-side shotgun. And with that design almost immediately gunmakers began using this action for double barrel rifles, or more simply double rifles. The advantage of having a second shot immediately available is useful in a huge number of situations, from fast moving upland game, to waterfowl, driven big game, and dangerous game. The disadvantage to this system is that it is incredibly expensive to produce, with double barrel guns having a large number of complex shapes, tight tolerances, and requiring strong steels that are hard to work with. There's also the added problem of barrel regulation, meaning having both barrels shoot to the same place, which is especially important on rifles since a deviation of a few inches can mean the difference between a game animal being killed outright and a game animal being wounded or lost.

The bolt action system in contrast to the double barrel system is much simpler to manufacture and much stronger than the break action system, and this makes them significantly cheaper to produce. Lever action, pump action, revolving, and self-loading arms fall somewhere in between, and have their advantages and disadvantages. There's also the single shot, which is extremely inexpensive but inferior in many regards.

When shooting birds over pointing dogs or shooting waterfowl with shotguns, target presentations were exceptionally fast. With a side-by-side or a self-loader, firing a second shot is as simple as pulling the trigger again. The difference between the two was that a self loader offers more than 2 shots, but the double offers two chokes, so it's up to the user which they choose to emphasize. The pump action is somewhat slower than the previous two, however a skilled shooter can operate the trombone action at speeds similar to the previous actions. Having more than two shots is an improvement over the side-by-side, and they were cheaper than both self-loaders and side-by-sides, so they became extremely popular with American shooters. The lever, bolt, and single action revolving systems are so markedly slower than these three other systems that they don't really offer anything over a single shot design and most couldn't justify the added expense. The throw on the Winchester 1887 was extremely long making it awkward and slow, and was more expensive than the later model 1893/1897 pump action designs offered later. Cocking the hammer on a revolving Colt 1855 shotgun is extremely slow. Bolt action repeating shotguns really weren't attempted until the mid 1900's.

The answer somewhat depends on when specifically you're referring to and in what context - military or commercial. It'd help if you could clarify since the specifics of the answer really depend on when and where you're specifically looking.

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In general though, we've got two sides of the issue that led to there being an absence of certain designs since the beginning of the 20th Century - military and commercial. I'll focus on military since that's where my expertise lies.

Military customers were generally very conservative and working with very specific doctrine in mind. A look at the decision-making for US, British, and French rifle design shows just how conservative military establishments could be - the British, for example, didn't remove magazine cutoffs from new production guns until midway through WW1. Military rifles needed things unnecessary for sporting arms, like bayonets, more powerful cartridges, longer barrels, and volley sights. Being procured in huge numbers, cost became a significant concern, especially in armies like the US, which would be a very small frontier force after the Civil War.

Actions considered unique today all generally ran into technical issues that precluded adoption. Lever-actions were consistently behind the curve when it came to being able to handle contemporary military cartridges - Winchester didn't have a lever-action capable of (safely) handling .45-70 until 1886, and by the time the Winchester 1895 was available to handle modern smokeless-powder cartridges, the US had already adopted a bolt-action rifle. Similarly, revolving rifles fell off over safety concerns (due to the cylinder gap) and weight (as stronger, heavier cylinders would be needed to withstand pressures of rifle cartridges). Double-barreled guns ran into similar problems of weight, effectively doubling the weight of the action for only a niche benefit.

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On the commercial side, a lot of decisions had to do with the market. A lot of the market consisted of rural people looking for something for hunting or something akin to a ranch gun. The added firepower of a repeater came with extra cost, and to many it wasn't worth it. Shotguns were hit doubly hard by this, as their utility is almost exclusively limited to sporting uses, where it's not much of an issue if you have to break open the gun to reload after every shot or two. Unusual layouts also tended to have some niche market that may not exist anymore. You brought up elephant guns as one example, but another market that was surprisingly prevalent around the turn of the century was "bicyclist" pistols - small pocket pistols in fairly anemic calibers marketed to city folk, ostensibly for self-defense against feral dogs. It's almost nonsensical to someone today, but they were a surprisingly common piece a little over a hundred years ago.