I'm not saying that they lose their right to an opinion because they are not paragons of virtue. I'm saying that these wealthy people
make no attempt at all to mitigate their environmental impact while telling you to essentially pay off their carbon bill. Elon Musk is not alone in taking his private jet on 10-30 minute flights for trips that could easily have been done in a car. Bill Gates has a gigantic unmitigated environmental footprint, for example.
Generally, you would expect people with more wealth and status to be focused on preserving that status. Part of that project is (ostensibly) preserving the world. If Bill Gates does not think that mitigating his own carbon footprint is important for the preservation of his fortune (including several seaside mansions, apparently), I doubt it's important for any of us in the preservation of our futures, despite what Bill Gates is saying about it. This is an example of the phenomenon of revealed preference (https://en.wikipedia.org/wiki/Revealed_preference) - you can learn a lot about what Bill Gates actually thinks by looking at what he does rather than what he says.
For the record though, I do think that what these wealthy people say is almost always the wrong thing to do in general. Wealthy people are known for giving bad advice on how to get wealthy, for example. Because some of them run their mouths a lot, they are also generally known for not being smarter than the average person outside of their area of expertise. Because I'm picking on Bill Gates, go ask anyone at the WHO or the CDC about their opinion of Gates' level of expertise on vaccines and disease prevention - it's not high.
Also, wealth and publicity are kind of correlated, but not that significantly. Bernard Arnault famously rarely says a word in public. Larry Ellison is very private. You probably haven't heard a word from Steve Ballmer, the Bettencourt family, or Larry/Sergei. The people who speak a lot about social causes are the ones who are trying to build a brand around those causes. You get a lot of famous actors worth single-digit millions (thanks to reckless spending habits) speaking about climate change, for example, while also not opting to adopt any green habits themselves. I'm picking on them too.
EDIT: I'm picking on rich people too much. You see the same behavior in politicians, too. They are completely unwilling to risk their political careers to make meaningful carbon impacts, and instead do inane things like banning plastic straws. That suggests to me that their level of concern is about at the level where someone would ban plastic straws: they want to do visible and silly things to placate the activists, but don't really think the issue is serious.