I thought it was Gay Nazis or something.

I’ve honestly not even heard Putin mention NATO on Ukraine.

One of the listed reasons, yes. Other reasons:

1. Protecting russia-speaking people of Donbass (almost identical reason Hitler gave for invading Austria and Sudetenlands in 1938)

2. Historical and cultural justifications, Putin claimed Ukraine is not a legitimate state (also an argument used by Hitler in 1938)

3. To stop “aggression of Ukraine in Luhansk and Donetsk” (Hitler accused Poland of the same in 1939)

4. Reclaiming rightful territory (Hitler referenced the Treaty of Versailles as reason for his wars)

5. Claiming that diplomacy failed and that Putin has no choice but to invade (also said by Hitler)

It’s almost as if Putin is Hitler reincarnated.