The extent in which Ukrainians want to "go their own way" varies drastically depending on the area, being stronger in the west and weaker in the disputed areas of the Donbas and Crimea, with ethnic Russian populations being a majority there, having deep links to Russia, culturally and ethnically. In a pre-2014 survey by the US Agency for International Development 40% of the respondents considered themselves Russians versus 24% considering themselves Ukrainian. 53% preferred Crimean autonomy versus 23% preferring joining Russia and merely 2% preferring being in Ukraine as a common oblast.
There is also the fact that Ukraine has a very nationalistic movement with intense anti-Russian sentiment, which in turns wants to destroy the cultural heritage of Russia inside the country. Case in point, the recently-assassinated activist and politician Iryna Farion, who among other things advocated for violence against Russian speakers and ethnic minorities. I want to say for the record that I am not justifying or condoning the invasion, let alone any atrocities omitted, but this overly simplified rhetoric of Ukrainian genocide, such as "Russifying" areas that had strong Russian influence and culture to begin with, kinda starts to fall apart.
Ukraine has used hospital and for military purposes according to Amnesty International but I do not believe this was the case here. In fact the New York Times reported only two civilian deaths, both adults. The missile used in the attack has a warhead with either 450 or 800kg of explosive material. A direct attack on the main hospital building would have leveled the entire building and buried dozens of people in the rubble, including children. All of the other missiles that were not intercepted by Ukrainian defenses hit strategic targets like factories and air defense. I only see two probably explanations. Either the hit on the hospital was a tragic accident or, despite all those alleged strikes in Syria, the Russians still haven't gotten the hang of how to conduct terror bombing.
Ever since WW2 it's been proven time and time again that terror bombing does not achieve the psychological effect to stop a war, it galvanizes the will of the civilian population to keep fighting. If you're going to be bombed in your home, if your hospitals are being destroyed and you're going to die of famine or disease even if the bombs don't get you, you might as well pick up a rifle and die fighting. It's counterproductive for the attacker. The hospital hit on Kyiv did motivate some Ukrainians to fight. Now the reports of civilian deaths in Kursk are also motivating the Russians to fight. It goes without saying that it is no different in the Middle East.