Is a Twitter account that uses this API to post all movements of a single private individual, or their vehicle really so unobjectionable, just because it comes from a public API?
At some point we will need either very strong social norms, or case law, or most likely legislation to address this issue.
Addressing the privacy impacts of programmatic operations on public data (i.e. what you can see from a window, or a satellite) is an important frontier for privacy and a largely unsettled question. You're just seeing it play out in this case because data for this particular vehicle type is public.
(Which, I should add, is surely just for legacy reasons and definitely a terrible idea. There's no public interest in being able to track everyone with a plane any more than there is a public interest in being able to see comparable data for anyone with a bicycle, car, or cellphone. Or look at it this way: when private flight become 100x cheaper and safer to the point where we fly instead of drive, do we want our movements tracked just because we were in the air? Of course not.)