Clearly he did not mean children could not pass on the infection.

This is why we don't trust random thoughts by duderinos on Twitter and it's why we have people who specialize in communications.

The error that Musk is making (and the one which BBC pointed out) was to downplay the fact that children can become seriously ill from this infection.

The fact that severe disease is much less prevalent in younger patients does not mean it is entirely absent.

Most people, even if they are intelligent, are not acting with their full logical faculties. We are busy, distracted, not always thinking, give to much credit, tired, read it poorly, not bothering, and many of us aren't so bright.

Which is why words matter.

Communication is a skill.

Musk is actually a talented communicator but in a special way, not the kind we want during a crisis where information is important.

Saying 'children are immune' is irresponsible, he should apologize and update the Tweet. To not do so is to fall into Trumpish communicating, which is bad.

You might want to update your internal definition of "immune".

Immunity implies that the host body has developed antibodies to neutralize viral replication or interferons which straight up kill viruses. As a result, the presence of the virus is reduced, possibly to near zero.

But that's not really the case here. You might be asymptomatic but still shedding virus just as much as someone on their deathbed.

With respect to children, they still become infected at the same rate. They still provide a place for viral replication even if they are asymptomatic.

Still, there are other less literal interpretations. You can be immune from legal action, or immune from the effects of a global financial crash. When Musk says:

...but elderly with existing conditions are vulnerable. Family gatherings with close contact between kids & grandparents probably most risky.

He clearly isn't saying kids cannot carry COVID-19. He's saying they're "essentially" immune from the symptoms.