I also never claimed any sources, I'm just giving my impression of him, which is that he's a bullshitter. But I can list a couple of things we do know about him:
At Tesla, despite not being a founder of the company, he contractually established that he was allowed to call himself a founder. And then pushed the actual founders out.
Online, we see him pick stupid internet fights, and post irresponsible tweets that got him slapped for stock manipulation.
At Twitter, we've all seen his bizarre mismanagement, despite his original background in software development.
We've recently heard that the Cybertruck was a bad decision that he personally pushed through at Tesla.
So given all of that, of course it's still possible that his rocket engineering creds at SpaceX are real, but you really can't blame people for having some doubts about that.
I don't expect anyone employed at SpaceX to spill the beans, but there's plenty of anonymous rumours on the internet about his actual role:
https://www.reddit.com/r/spacex/comments/27rnwj/is_elon_pers...
> Elon is indeed very involved in the design. The way they explained it is they give him a list of options to choose from and he picks the one he likes. Usually, it's the one the lead engineer wants, but sometimes it isn't, and Elon gets what he wants.
That's reasonable enough for a CEO in a flat organisation, but it doesn't sound like he's doing the actual engineering, just picking from the options the engineers give him.
I've also seen a (claimed, not verified) SpaceX employee say that SpaceX has people who's job it is to keep Elon happy and feel involved, because a happy Elon gives them the freedom to make the right decisions.
He got this Tony Stark image in the media, and I think he's been leaning a bit too hard into it, and started to believe the image that he could do everything. And the history of Tesla shows that he's not above overstating his own role.
Don't get me wrong: I love much of what he's done at Tesla and SpaceX; EVs wouldn't be where they are now without him, and rocketry in the US might well be dead without SpaceX. But he certainly has his share of character flaws as well.