I would argue that in the time frame listed above...into the 50's...ammo manufacturers really picked up their game in the post war years and gun manufactures followed.
Now...Winchester Model 21 introduced the three inch 20 gauge shell that was made available to the public in 1954 as the "Western Super-X Magnum.
So, with that, the public was introduced to a 20 ga. shell that could carry the same loads as the standard 16 ga. stuck in the 2 3/4" world. Most people do not concern themselves with bore diameter/pattern efficiency with different shot sizes or velocities or whatever. To most 1 1/4 oz of shot is the same regardless of being spat from a 2 3/4" 16 ga. or a 3" 20 ga.
In that time period the 16 simple got squeezed out between the heavy 20 and light 12 ga. loads. As far as European influence, my reading says the the 16 was made most popular in Germany. Ironically for the same reasons I (and all other 16 lovers) are fond of saying: Carries like a 20...hits like a 12.
You could make an argument the 16 might be being similarly 'squeezed' today. I'm very fond these days of carrying a light little 20 SxS with 1 1/4 oz of 3" hevishot #6 shot. Deadly and still lighter than my Sweet 16 or my BPS.