I don't think it was intended as a Nazi salute and neither does the Anti-Defamation league:

"the Anti-Defamation League called it “an awkward gesture in a moment of enthusiasm, not a Nazi salute.”

But we hold public figures and those particularly that work in the media to different standards, right or wrong.

A person like Musk has to be more careful not to give the appearance of an offensive salute.

People who work in the media have to be careful about who they malign because it is often considered an expression of the medium/company itself and not just its employee.

If you own a radio or tv station, you are justly concerned about what your personalities espouse and publish as it is often taken as a reflection of the stance / opinion of the station itself.

This is a matter of employment / contract law, not so much first amendment law.

It is a legitimate concern of an employer. And the newscaster who goes around in public giving the Nazi salute would find themselves losing their job just as quickly as if they were to attack someone else for doing the same.