I think there is a few factors at play here.
Firstly its a misconception that that VLS has anything to do with this. VLS has only really been mounted on ships from the 80s onwards. The first US ship with VLS was the Ticonderoga class cruiser launched first in 1980, however the VLS was only used for anti aircraft missiles and the Harpoon anti ship missiles were positioned in quad launchers at the rear of the ship. The Soviets did in fact deploy VLS earlier than NATO on the Azov class cruiser, but kept external anti ship / submarine missiles on many designs. Indeed throughout NATO it was common to mount anti ship weapons externally just as the Soviets did right up until the mid 80s. If you think about it, anti ship missiles want to go sideways rather than upwards so it makes sense to point them at where the enemy will be.
Given that the majority of anti ship missiles were external on both sides I suspect the difference in prominence and perception comes from two linked points.
1: The majority of NATO ships were not designed as surface combatants, rather as escorts for carriers
2: Soviet anti ship missiles tended to be bigger with higher speeds and larger warheads.
On the first point the main priority of the US navy is power projection. As such it is generally reliant on carriers to provide strike capability through sorties. This is suplemented by submarines for anti shipping capbilities. THe majority of NATO surface combatants are geared towards allowing those carriers to be deployed unimpeded, as such they have a lot of anti air and anti missile capbility which by nature fires upwards.
The Soviet navy was more focused on area denial as they did not develop a significant carrier force. As strike capbility was not provided by naval aircraft (at least to the same extent) a greater emphasis was put on sinking enemny combatants. As such their ships were designed for surface to surface roles to contest the baltic and black sea. This required them to be able to sink NATO carrier groups themselves which in turn required more powerful missiles with higher speeds and bigger warheads. These therefore which appear far more prominently on their ships. I'd also note that the Soviets tended to put large missiles on smaller ships such as corvettes which made them appear even more prominent.
Russian missile systems were during the cold war (and perhaps still are) superior in many ways to NATO counterparts. The USSR had no problem devloping reliable rocket motors and had some pretty fearsome weapons. I think the idea that they put their missiles on the side to prevent self damage is a little far fetched.