So, Picard, career diplomat that he is misread the room. The scary part and probably why Picard beat himself up so much is that he probably COULD have done a lot more by using his oratory skills to persuade people to help versus actually doing the day to day organizing. He could have done a lot as the "face" of the evacuation both pre and post attack. However, he's so gobsmacked he's unable to do anything before the Sun explodes (and there was barely any time left anyway).
I guess that's really the part that I have a hard time wrapping my mind around. Picard is usually very gifted and reading things like that, and he has a way of convincing people. And it wasn't like the warning signs weren't presented. In fact, IIRC there were times his superiors almost begged him to do just that. And he just blew them off. He had a job to do and anyone who had any concerns just had to suck it up buttercup. And that actually made it worse.
The worse thing you can do is tell someone who has concerns is 'talk to the hand' basically. Your basically are showing them that they are not important at all.
But again I guess that's part of the theme of the show. Picard is thunderstruck that the Federation abandoned the evacuation. But also that there is probably some self-blame as well. If he had done some of those things maybe he could have salvaged something from the situation. I haven't seen the show but it sounds like they haven't solved the mystery of what happened to cause the androids on Mars to suddenly immolate the entire planet. So I imagine he might be wondering if there was something he missed on that front as well.