I feel like lever actioms shoot faster and carry more bullets than the Springfields. Winchester was also a pretty big company (I think)
The lack of lever actions in frontline service prior to WW1 by most Great Powers is due to a variety of factors largely relating to doctrine. The US in particular had a lot of driving reasons that paralleled concerns of most other powers, but exacerbated by the country's unique circumstances between the Civil War and WW1.
In general, armies all over the world were reluctant to adopt magazine rifles en masse owing to concerns over ammunition expenditure. In an era where supply of armies was generally very precarious and strained on campaigns, there was a very real fear of excessive ammunition expenditure from soldiers firing in a panic would heavily strain logistics and even lead to critical ammunition shortages. Other concerns endemic to magazine rifles of the time included issues like per-unit cost and ammunition type (Winchesters in particular weren't capable of chambering military rifle cartridges for some time). For the US in particular, these issues were a bit more pressing than for the continental armies of the Great Powers in Europe. After the Civil War, the US Army was a comparatively small force with a very limited budget, and was almost exclusively fighting on the frontier, where supply was even more strained than normal. American doctrine around this time would come to stress the marksmanship of the individual soldier at comparatively long distances, a trend that would continue in some shape or form all the way to the early Cold War. The standard infantry arm until the adoption of the Krag Jorgensen in 1892 was the Trapdoor Springfield, which was adopted in large part thanks to its simplicity and the fact that existing stocks of muzzle-loading rifles could be easily converted to the new design. And then, to top it all off, the .45-70 cartridge that was the standard of the time was too powerful for any existing lever-action rifle until the Winchester 1886 - at which point smokeless powder cartridges made the .45-70 round and guns that fired it obsolete.
So when we look at lever-action rifles in the pre-smokeless powder era, we see that they're practically the antithesis of everything the US Army needs. They're expensive guns chambered in weaker, shorter-ranged cartridges, and their chief benefit - rapid fire - was seen as a liability in the kind of fighting the US Army was expecting. And even their rapid fire capabilities weren't all that important in a long-term engagement - in an era before the advent of rapid-loading devices like the stripper clip, detachable magazine, and en-bloc clip, magazine rifles had no rate of fire advantage over a single-shot in a long-term engagement. Magazine rifles like a Winchester may get more rounds off in succession at the start of an engagement, but once the magazine is expended, the soldier is going to spend significantly longer reloading the magazine, while the a single-shot rifle will be keeping up the same consistent rate of fire.
The US did finally warm up to the idea of a magazine rifle soon after the advent of smokeless powder in 1886, but a lot of the concerns that drove procurement to that point lingered. The US Army was still concerned over excessive ammunition expenditure and (as with many other armies) required a magazine cutoff on their rifles so that soldiers would use the rifles as single-loaders until otherwise directed by their commanders. A modern small-bore smokeless powder rifle was also wanted, which would materialize as the 30-40 Krag. Among other factors, the ultimate result was that the Krag Jorgensen fit US requirements fairly well, while the only competing lever-action design (a precursor to the Savage Model 1895) was deemed too complex for adoption. Although the failings of the Krag Jorgensen made apparent in the Spanish-American War would lead to the adoption of the Springfield 1903, the second round of rifle trials wouldn't give lever-actions any room to compete. The US Army was heavily influenced by the Spanish Mauser Model 1893s they encountered in the war, and from the start would copy a lot of Mauser features as they developed the new rifle. A more powerful cartridge was developed (30-06), a Mauser bolt and magazine system was used, and a magazine cutoff was added to the design.
Even when the US entered the war and rapidly had to re-arm faster than existing arsenals could handle, they opted to rework the British Pattern 1914 design for 30-06 to become the Model 1917. Thanks to the ease of converting existing production lines, US commercial arsenals (Winchester included) were able to churn out so many 1917s that the gun was more common in US service by war's end than the "standard" 1903 it was supposedly a substitute for.
Winchester did make lever-action rifles that were used in WW1, but not by America. Britain and France both had placed small orders for more traditional Winchester rifles through the war, and, much more famously, the Russians ordered a substantial number of Winchester 1895s that would see significant service in the war.
British purchases were aimed at arming troops in secondary roles, and, like most of their non-standard rifles they would procure during the war, they were purchased with the intention of freeing up rifles from the front. These orders would amount to:
France would also procure small numbers of similar patterns for similar uses, although I unfortunately don't have any hard numbers for it.
The Winchester 1895 was the only realistic option for a lever-action service rifle by WW1, as it was the only commercially available option that could really mount all the necessary military features. Winchester would chamber the gun in a variety of different calibers, but for Russian service they created a variant that chambered the Russian 7.62x54r rifle cartridge and could use the same stripper clips as the Mosin-Nagant. Features to make it more suited for military use like full-length stocks, handguards, adjustable sights, and a bayonet were also incorporated. However, while the Winchester 1895 served reasonably well in Russian service and into the Spanish Civil War, it only ever took this form at the outbreak of WW1, with the Russians desperate for rifles of any type and Winchester looking to bolster sales of a gun that had thus far struggled to find a good market.
While /u/Meesus has given an excellent answer, it's also worth taking into account that Spitzer bullets had been developed in the early 20th century. These gave substantially improved accuracy due to the more aerodynamic shape of the bullet. However, the problem with Spitzer bullets is that the sharp tip of their point can be used to ignite the primer of another bullet that it is pressed up against. Lever-action rifles use a Tube Chamber, which means that all the bullets are in a row with the primer on one bullet in direct contact with the nose of the preceeding one. Dropping a tube chamber rifle, like the Winchester, when it's loaded with Spitzer bullets is just asking for a negligent discharge inside the tube chamber which will destroy the weapon and also probably hit someone on your own side.
Bolt-action rifles, on the other hand, enable a magazine where the rounds are stacked vertically, rather than horizontally. This means that the primer of each round is not in contact with anything until it is hit by the firing pin when you actually want to shoot the rifle.
I'm not a gun expert so it's possible that there exists a rifle that was lever-action and overcame this limitation, by doing away with the tube chamber, but I expect that such a rifle, if it did exist, was prohibitively expensive (or deficient in some other form).
The majority of the technical reason behind the adoption of magazines was to fire this superior Spitzer round, and as this corresponds with the doctrine it was simply a no-brainer.