"Staged combustion (Замкнутая схема) was first proposed by Alexey Isaev in 1949. The first staged combustion engine was the S1.5400 (11D33) used in the Soviet planetary rocket, designed by Melnikov, a former assistant to Isaev.[1] About the same time (1959), Nikolai Kuznetsov began work on the closed cycle engine NK-9 for Korolev's orbital ICBM, GR-1. Kuznetsov later evolved that design into the NK-15 and NK-33 engines for the unsuccessful Lunar N1 rocket. The non-cryogenic N2O4/UDMH engine RD-253 using staged combustion was developed by Valentin Glushko circa 1963 for the Proton rocket."
And way down at the bottom of a long engine list spanning more than half a century...
"Raptor—SpaceX LCH4/LOX engine in development, first flown in 2019[19][20]"
(Yes, there are others listed, but SpaceX was first to actually fly something.)
That the IPD was hydrolox and didn't proceed to the point of a full engine, let alone a flight is besides the point.
What you said was: "They kind of did invent full-flow engines though", which is completely false.
They kind of did invent full-flow engines though, which were an unattained holy grail of rocketry.