I think operating a bolt is more of a learned thing than operating a pump or a lever. Most bolt action shooters drop the rifle to waist level or at least down off the shoulder to cycle the bolt. A trained man on a bolt action is a thing apart. While at a significant distance for timed fire, aiming will always take up more time than working the action, I still find the lever much faster than the bolt for getting hits at close range.
I didn't start off liking pumps. The forend on the new Remingtons always seemed to flop around and that irritated me. It would work I suppose in the woods at close range but at a cost I didn't want to pay. I didn't learn what a pump could be until I put my hands on a Colt Lightening 38-40 made in 1896. The forend lacked that side to side flop, and no way could I cycle a lever anywhere near as fast, or remain on target as well. The Remington Model 25 from the 20's is almost as good. I doubt I could equal the rate of accurate fire at close range I can get from the old Colt with a semi-auto of similar power and weight.
As far as why pumps are rare, that's simply a matter of the original popularity of the Winchester lever. Winchester built rifles and Colt built pistols. When Colt began encroaching too much on the rifle market, Winchester showed them a model for a revolver asking for production advice, which amounted to a veiled threat. An agreement came out of that. It had nothing to do with the inferiority of the pump. Before World War II pumps by Remington and Marlin never touched the dominance Winchester held in the market. And of course after World War I, the returning GI's wanted bolt guns.
I can't really speak to the reliability of the early pump rifles the way I can the bolts or the levers. If I need a part for a Winchester lever or any of the common bolt guns, I can find it, although I don't recall ever doing that except for rifles that were abused that I have put back in service. The early pumps were rarer, and I would have a harder time finding replacement parts. For that reason I tend to be more careful about the conditions in which I would subject an old pump.