Not everything. Picard was command track his whole career, I think the same was true for Sisko. But Janeway started as a science officer before switching to command, and Archer was a test pilot. Their experience in those fields would sometimes be useful. It's similar to how Picard's interest in archeology was sometimes useful.
I imagine they'd also conclude Picard is a loser.
The main issue was his PTSD.
I don't like the premise that Picard, alone, out of the entirety of the Federation, is the only thing standing between the Borg and the assimilation of Earth. It seems to go against the collectivist identity of the franchise. It's fiction, they could have written it another way, they just didn't because Picard is the star of the show.
And the PTSD isn't even the issue, he was compromised by alien mind control implants from a species far stronger than the Federation, with hostile intent. Implants the Federation couldn't even remove. Implants which have the property of aggressive assimilation. A single nanoprobe is a potential extinction level event.
Humans lose their security clearance by smoking a joint. Being puppeted and mind-controlled by eldritch alien gods into a system-wide genocide should at least lead to a demotion.
And everyone forgets Geordi was tortured and mind-controlled by the Romulans. In anything close to reality he'd never be allowed on a starship again, and Picard would spend the rest of his life quarantined inside a force field getting scanned like that guy in the X Files with the dark matter shadow.