I understand what you're saying that NATO isn't a threat to Russia, but Russia thinks it is.
However I don't think that's true - after all, Putin is not justifying his invasion with NATO back home, he's come up with this entire fictional genocide on ethnic Russians and how the country needs to be liberated from Neo-Nazis.
That kind of lie is incredibly difficult to pull off. He would have invested billions in propaganda and misinformation and social media trolls, and enforcing restrictions on the press.
If the Russian public would have accepted NATO threatening Russia as an excuse, he'd have just used that.