“little different to the Vatican as far as giving its stamp of approval to the state”
The Vatican, unlike the Russian Orthodox Church, has had to deal with a great many states at once for over 1500 years, ever since the collapse of the Roman empire. And if you dig into Church history, there was perpetual conflict with these potentates. If it wasn’t one character acting up it was another. In the medieval period there was always some king excommunicated or threatened with it. That’s one reason the Church was so desperate to hold on to its own territories in Italy, the Papal States.
All these kings and dukes and whatnot wanted church revenues, church lands, control over appointments, etc. and etc. Often enough they wanted control over Popes.
This famous incident gives some idea of what the Church was dealing with on a regular basis -
https://en.wikipedia.org/wiki/Road_to_Canossa
And whenever some such situation was sorted out, there was another.
The Vatican crowned numerous heads of the Holy Roman Empire. Intimstely involved in machinations in Europe. The Catholic faith was the official religion of the HRE. Akin to EOC in Russia
And let’s not forgot the periods when there were 2 popes, one in French territory, and the other in “Italian” territories. Lots of political and even armed struggle to hold on to one or the other by different national/regional groups. One book which give some info is Barbara Tuckman’s A Distant Mirror about 14th century European difficulties. Both churches have had to deal with periods of intense survival threats. Certainly surviving Putin is one, surviving Hitler and Stalin another.