Probably its a cultural thing. In WW II Western civilization countries viewed sniping as a cowardly way to fight a war, practiced by those without the wherewithal to go head to head, Mano a Mano.
Given this, one can only conclude that Russian troops had a mind numbingly huge number of weakly countenanced, wimps and cowards within their ranks.
^&confuse^&confuse^&confuse
If one is to follow that same train of thought then the US would be wimps and cowards since Korea forward- not sure that logic fits. The Russian soldier is as formidable as any on the planet- if anything it's their leadership that is highly suspect- but the actual soldier is as lethal as any that ever existed.
The Western allies did not incorporate "Snipers" in their Order of Battle (OOB) in any meaningful numbers because snipers really do not fit what the Western Front was in 1945- a rapid drive to Berlin. Ground was to be gained as quickly and as efficiently as possible. Snipers do not fit that mold. Snipers are better used in a defensive position and essentially act as a force multiplier allowing them to exact precision kill capacity against invading targets high value assets like senior leadership, disrupt communications or other mission essentially tasks. At a squad or company level, they are also used by modern forces to provide covering fire during a host of different missions.
The Fins knew they were going to be overrun by the Soviet juggernaut, it was just a matter of time. Amongst dozens of other defensive measures they took (like flooding fields, cutting lines, blowing bridges, etc), they did an expert job in the deployment of their snipers. In many ways, the Soviets learned a very costly lesson that they were able to leverage a few years later in Stalingrad and other areas where the Germans had them under siege. The Fins took advantage of their hunting culture and their innate knowledge of their home country and exacted an incredibly lethal kill tally against a Soviet Army that was utterly unprepared to fight that fight. Any army that takes to another country fully understands that playing on their field gives them the home field advantage which the Finns clearly had.
When Barbarossa stalled, the German forces settled in and adopted a siege rather than advance and attack strategy. As such, this gave the Soviets time to properly deploy their sniper kill teams similar to how the Finns did in 39. The soviet snipers were all very similar to the Finn snipers- mostly from rural hunting communities with excellent marksmanship. This time, the Soviets were playing on their ground, the Germans were the aggressors and unfamiliar with snipers being deployed in such high numbers. As such, the Soviet sniper teams were able to rack up large body counts- just like the Finns did to them a few years earlier.
The US military (and our Western allies like Canada, Australia and England) uses snipers in a variety of roles from Seal Team 6 picking off Somali pirates, providing overwatch during infiltration patrols in Iraq, and taking out high value AQ leadership assets in Afghanistan. Still, snipers were not used as a viable military option until we began the rebuild process in Iraq. When we were rolling through Iraq in 2003, we adopted an offensive position designed to take ground. Snipers were not actively involved with offensive operations simply because they do not fill that roll. Once we took Iraq (and became immeshed in Vietnam), we took a "Defensive" position and the sniper could finally do his thing. Soviet snipers are still being used in the Ukraine (and yes, there are female snipers in those units as well).
So to answer the original question, snipers were never deployed in any meaningful numbers in any stage of WW2 by the Western allies simply because we were taking ground and constantly on the offensive. Snipers are best deployed as A- part of a support to patrols during pacification or B- as a precision asset with a mission with a very singular purpose.
Hope that helps
CC