No one was about to invade Russia, not today. But NATO is by definition an alliance created against Russia, and it has a long history of trying to initiate regime changes in its enemies.
You have to remember that Russia is a federation, one with much deeper internal divides than the USA. The closer we are to Russia, the easier it is for us to fund, train, and arm Russia's separatist elements.
If NATO was officially present if Georgia, do you think it would ignore any pleas for help from chechen separatists trying to defend themselves from the butcher running them? Or would we actively work to try to make Chechenia an independent country, over the next 10-20 years?
And Chechenia is far from the only vulnerable place.
Note that I am not saying this would be a bad thing for the people of Chechenia. It is quite likely what a lot of them want. But if Russia were funding, arming, and training the IRA, or Texas's secessionist fringe, would the USA or UK be happy to allow this? No, empires are fundamentally built out of people who try to maintain their empire.