Why did bolt action rifles become the standard military rifle instead of lever actions?

You're holding onto a few misconceptions here. The biggest issue relating to the use of lever-actions was the strength of the action - available designs weren't capable of withstanding the forces produced by standard military cartridges really up until everyone had adopted a bolt-action rifle in the 1890s.

As you allude to in your question, these rifles typically had shorter ranges than the standard military rifle cartridges they were contemporary with, and that was a consequence of them using weaker cartridges. If we go all the way back to the Henry, it's pretty clear just how much of a liability this would be at a large scale. The Henry's cartridge was notoriously weak and short-ranged, with an effective range by modern standards reaching only to about 150 yards. In an era where standard rifle sights went out several hundred yards and armies were training men to fire volleys on area targets out to these distances, this was quite an unacceptable drop in performance. Later lever-action designs fortunately wouldn't be so limited in their capabilities, but still they were generally limited to cartridges with less performance than the standard military cartridges of the day. Winchester did submit a .45-70-chambered Model 1876 to US Army trials, but the design was passed over and the action really wasn't up to the kinds of pressures produced by that round. They did finally come out with a model in 1886 that was capable of handling pressures in those ranges, but at that point it was all irrelevant - smokeless powder had come along and made lower-pressure black powder cartridges obsolete.

Only by the early 1890s did lever action designs come along capable of handling smokeless powder cartridges - the Savage and Winchester models 1899 and 1895 (respectively). Both had earlier incarnations submitted to ~~US Army trials~~ (losing to the Krag Jorgenson) **correction - after looking over my reference books, it was actually New York State National Guard trials**, and both were notable in that they lacked the tube magazines earlier designs were famous for - instead opting for a rotary magazine (Savage) or vertical box magazine (Winchester). Savage would have limited success with military contracts to the New York National Guard and Quebec security forces in WW1, but only Winchester's 1895 would see any significant military use, with about 300,000 made for Imperial Russia during WW1, where they'd perform reasonably well.

However, I'd contest the claim about the guns being any faster firing than any other repeater. As much as movies and pop culture tend to make lever-actions look like they're quick and easy to cycle, most leverguns aren't that smooth to operate, and more robust models like the Winchester 1895 and Savage 1899s had a significantly less smooth action. On top of that, there have been plenty of bolt-action rifles that are famous for their smooth actions, like the Krag, Enfield, and Schmidt-Rubin systems, all of which offer comparable rates of fire to a "fast-firing" lever gun when both are in ideal conditions.

On a more practical note, fire rates were less important than you might expect. Really right up until WW1, armies were more concerned over excessive expenditure of ammunition by poorly disciplined troops than they were about fire rates, as logistical constraints were expected to be a huge issue. Up into the first years of WW1 armies were producing rifles with magazine cutoffs, the idea being that soldiers were to single-load ammunition until otherwise directed by an officer. This mindset went back some time, and the presence of magazines on rifles at all was something that didn't really gain traction among the Great Powers until the 1880s (for the same reasons stated above).

That being said, even ignoring the doctrinal concerns, the effective fire rate of a traditional lever-action gun is about the same as a single-shot. Although the lever-action may be able to fire without loading until its magazine is empty, the shooter then has to single-load the magazine to refill it. This in and of itself is one of several reasons that tube magazines fell out of favor for (generally lower capacity) box magazines - box magazines were easier to design speed-loading devices like stripper/charger clips and en-bloc clips.