The GDPR only applies to businesses, not individuals. So, clearly in this case it does not apply at all since the person behind @ElonJet is acting in an individual capacity.
The GDPR applies to everyone, except "by a natural person in the course of a purely personal or household activity", member states and some authorities.
I can go on one of those websites, see all those flights, but I have no idea who they belong to or who's on board.
That changes once someone specifically links a private jet to someone.
Within the EU I'd be interested in the ramifications with regards to the GDPR.