Yes, because it's not really public vs. private that matters, in both cases it's whether the shareholders care. (If it's public they complain to the SEC, if it's private they complain to their contract lawyers who hopefully included some accountability in the contracts.)

This is also why Musk can get away with it - he's got fanboys, albeit of slightly different types, dominating the shareholders in both cases. (In the case of Twitter he may even be personally the majority shareholder with no obligations beyond cash now, but who knows...)

In this case, isn’t Elon the only shareholder?

I'm not really sure - I think that depends on the details of how X Holdings is set up, which I'm not going to bother looking into. I should've used a more general term like "creditor", but the point remains.

We don't have really any visibility into what short- and long-term obligations Musk owes the various financing parties.

Edit: https://www.reuters.com/markets/deals/saudis-kingdom-holding... says Saud is keeping their ownership stake post-merger (they just love free speech!), so I do not think Musk is a 100% shareholder.