The integration into the NATO missile defense network is not a threat to a neighbor (Russia). The whole point of the missiles in that integrated system is defense, not attack.
I think the Russians have argued that they see it as a potential threat to their ability to use ICBMs and so on.
As to NATO expansion being seen as a threat, I can see that in a strategic context if you are a Power viewing other Powers. Understood.
AFAIK, Russian authorities tend to view foreign politics as a zero-sum game.
For my money, one of the great political opportunities missed was the chance to try and talk Russia into eventually joining NATO after the wall fell.
I don't know about the Yeltsin period. IIRC, Putin later rejected the idea of applying for membership, thinking that Russia should be invited.