In practice, humans are neither entirely rational nor entirely irrational. Humans occupy different positions on this spectrum according to genetics, culture and circumstance.
If we want to avoid triggering escalation events that may lead to an eventual nuclear episode it seems sensible to avoid certain emotionally-charged and chaotic situations such as having the armed forces of nuclear-armed states or their close allies in direct combat with each other.
That's true, yes. But in an alternate reality, where NATO doesn't exist after Cold War, the European countries have built up their own defense (maybe shared to an extent to allow compatibility), and they create an ad-hoc pact based on situation presented, this would, I believe, give us better options to help defend Ukraine. And maybe, who knows, Putin wouldn't even go crazy being paranoid about NATO.
There is no reason for Russia to do so. Ukraine is not part of Russia. There already is a precedent for military operations in a country occupied by Russians (when Trump bombed Syria).
That would trigger a nuclear bomb from Russia, the same as with NATO's involvement. When a nuclear power gets involved (and an EU army would posses nuclear weapons) then we enter the MAD scenario even if we want it or not. Re-labelling the Europe-based part of the NATO forces as "EU Army" I'm afraid won't foul the Russians.