I don’t know how sure we can be that Putin would have invaded regardless of assurances if NATO etc. I think he is using the Ukraine as Russia’s rightful land to get Russian citizens on board more than an actual justification to invade. This has a lot to do with the geopolitical makeup of NATO’s relationship with ex-soviet states, and Russia wanting a clear distinction between them. Again, I just think more could have been done.
The reason Eastern European countries want to join NATO is purely down to Russia continuing to interfere in their affairs. Its not fair really to just deny Ukraine membership of NATO or the EU, as this war is entirely Russias doing.
Since Ukraine became independent, Russia has poisoned presidents, rigged elections, annexed Crimea, armed and supported separatists for 8+ years. If they had just left Ukraine alone none of this would be happening.
Putin is a paranoid despot who thinks NATO is going to one day invade Russia, which is of course ridiculous. He has deliberately interfered in the West by supporting things like Brexit and targetting the US elections through propaganda bots. HE has bought this on himself