Your analysis is flawed. The media coverage had very little if anything to do with the reason people stayed. I was there I was in Ukraine on the day of the invasion. I have many friends and every single one of them was well aware of what might happen in the weeks and months leading up to that day. Ukrainians aren’t morons they have been well aware of the threat from Russia for almost a decade. For people in the US the war started last month, but for Ukrainians well they have been at war with Russia since 2014. Invasion by Russia has been not just a possibility for the last 8 years but a daily fact of existence.
The real reason people stayed in Ukraine is very simple. It’s their home. It’s why many of my friends who could leave if they wanted to are still there. I’ve had countless conversations about this with them. It’s not as easy as people make it out to be to just leave and go to another country. Even if you have the money there are considerations beyond money regarding uprooting your life and moving to another country never knowing if you will return. These people didn’t leave because they had hope that things would be ok, they didn’t leave because these are their homes. It wasn’t because of what was or wasn’t being reported in the news.
Ukrainians aren’t idiots they are far more media savvy than most western countries as they have been dealing with Russian propaganda for decades.