I am not sure it is a lie TBH, it is probably the only "reason" that may have, from Putin's view, at least some coherence. However it is mostly a flawed argument, since NATO has not invaded any country. In fact, many countries did not support US stance on Iraq other than US a a few lightweights.
I am still wondering if there is out there a system of early nuclear interception that works in a 300 or so km range from the launch point. That could explain all the fuss about security.
Fair enough, it may be not a lie in the sense that Putin et al may honestly believe it, but the fact is that no one wants to fight Russia. A resource-rich country with nuclear weapons... it's in everyone's best interest for it to be stable and friendly, so much so that even in the face of aggression (like 2008 in Georgia and 2014 in Ukraine) the rest of the world was like "nah, Putin's actually a good guy and if we buy enough gas from him, everything will be fine".
As for interception... considering that NATO deployed Patriots only in Romania (AFAIK) in the 20+ years since the expansion eastward began, and it was Romanians buying it, not NATO "donating" it, doesn't seem like it was a huge concern, at least not geographically. Now probably NATO will stack missile defence three rows deep along the entire border, again a great victory for the geopolitical genius Vladimir Vladimirovich.
NATO's strategy is to get closer to Russia and thus reduce the reaction time for the Russian missile defense system. Russia's strategy is to have a significant advantage in the means of delivering nuclear weapons, by increasing the range and speed of missiles. NATO and Russia are both quite successful in their strategies, but whose strategy is more effective I hope I will never know in practice.
In simple words, Russian missiles are now so fast and long-range that NATO does not have adequate countermeasures. They can fly over the North Pole, bypassing NATO missile defense systems, they can even fly over the South Pole. This is if we are talking about land-based missiles, and Russia's nuclear triad also includes sea- and air-based missiles, which are on constant combat duty.