Um... where the hell did you get the idea that the Russians use gas generators (inefficient) and the US uses staged combustion? That is almost perfectly backward.
Staged combustion was invented by a Russian, Aleksei Mihailovich Isaev.
The first staged combustion rocket engine built was the Soviet S1.5400, first flown in 1960.
The (ill-fated) Soviet N1 moon rocket used staged-combustion NK-15 and NK-33 rocket engines (the American Saturn V moon rocket used gas generator rocket engines).
The first western (German, not US) staged combustion engine was in 1963, and it was a laboratory test only.
The Russian Proton rocket family was using the staged combustion RD-253 rocket engine in 1965.
The US buys staged combustion RD-180 engines from Russia for United Launch Alliance's workhorse Atlas rocket family.
As far as I can tell, the first US-built staged combustion rocket to fly was the RS-25, better known as the SSME (Space Shuttle Main Engine), which first flew in 1981. It was a fuel-rich staged combustion cycle, made possible by the use of non-coking H2 fuel. However, by that point the Russians had been using oxidizer-rich staged combustion (which requires advanced metallurgy that the US could not duplicate for over two decades.
Now, both SpaceX and Blue Origin are US companies working on staged combustion rockets, but those are recent projects. In SpaceX's case, it is a full-flow staged combustion rocket, which is extremely tricky; no FFSC rocket has ever flown, although the Russians built and test-fired the RD-270 in the late 60s. SpaceX's Raptor has successfully fired on a test stand, the first FFSC rocket engine to do so since 1970 and the only US-built one to do so ever. The US (through private contractors Rocketdyne and Aerojet) experimented with FFSC in the "Integrated Powerhead Demonstrator", which wasn't even a full rocket motor; the front-end ("Powerhead") component was tested at full capacity in 2006, but then canceled; no full rocket engine was ever built using that design.
So yeah, the US historically didn't have shit on the Russians when it came to advanced rocket combustion cycles. That may be changing now, but it's driven primarily by private industry.