The rich have both security and privacy concerns that make regular air travel problematic. Bill Gates sitting in an airport terminal is a danger to him, those around him, and is also a hassle to him and those around him.

There is the secondary factor of their time being too valuable as well, and having a direct flight to anywhere in the world making a big difference in how much time they spend.

The core problem is that you're making it harder to be an environment supporter. It's the equivalent telling the average Joe that if he supports the environment but still eats meat daily he's a hypocrite and should feel bad. The environmental impact of raising meat is well known, so advocating for environmental measures while not taking the trivial effort of not eating meat one day per week is pretty hypocritical as well. I'm not sure about you, but I'd rather have as broad support base as possible to get environmental measures promoted and passed, than a smaller group of diehard activists each trying to police each other on ideological purity.

Would it be better if he just shut up and flew his private jet? That is, would it do less harm to the environment?

You are right about the argument in itself. It is poor. But the argument is not in meat of the argument, but in optics. The perception of things as 'rules for me but not for thee' almost immediately overrules even basic logic. In other words, make pain even across the board and people might listen.

You're speaking from the perspective of someone for whom the virtue of making personal sacrifices for environmental reasons is already obvious. Try putting yourself in the shoes of someone who doubts this. To those people, rich jetset climate activists come across like politicians who agitate against gay people then get caught having anonymous gay sex in airport bathrooms. Their hypocrisy is seen to invalidate the authenticity of their message.