It's simplistic (wrong actually...) to say that NATO, and the EU were being used by the US and western European states as an anti-Russian tool - there was a very short period when Russian accession to NATO was being talked about, but it lasted less than a year, and it tailed off because Russia wanted to be treated as the big dog in NATO east of Berlin, with a Russian veto over where US forces could go, who the eastern states could buy their gear from. It was all very 'Spheres of Influence', and Russia got very arsey and threw it's toys out of the pram when it was made clear that that wasn't how Russian membership was going to go.
It was the eastern states who banged on NATO's, and the EU's door, not the other way round - and it was because, for all the lefty talk about the fall of the Soviet Union, for the eastern states it was simply window dressing, it was about Russia, meet the new Russia, same as the old Russia.
They knew, felt it in their bones, that in the end Russia would be back - 5 years, 10 years, 30 years, it didn't matter - Russia that doesn't really believe them to be proper countries, doesn't believe them to be equals, places that exist to act as buffer zones for Russia, places for Russia to exercise more or less control over as the situation demanded.
NATO has become anti-Russian organisation, but In only in the same way that your front door is an anti-burgler device...
Amusingly, NATO has long held off acting as it's eastern states would wish because it sort to avoid pushing Russia into Chinas bossom as a client state - there was a widespread maxim, until quite recently, that Russia had perhaps 10 years before it would be forced to decide whether it wanted to be a European state, or a Chinese vassal state. NATO hoped it would decide to be a European state, not least because that would keep the Chinese somr distance from NATO's border.
What will happen now, with Putin backed into a corner with the economy tanking, and China seemingly not overly pleased with their partners actions, is anyone's guess...