As far as I'm concerned, being an individual that experiences life in a relatively weak gravity field whilst never moving more than a tiny fraction of C relative to anything that is of interest to me , I do NOT see the need to invoke GR when dealing with every day mechanics.
Accordingly, Newtonian physics is perfectly adequate for teaching PofF. Introducing more advanced physics simply adds confusion to a student cohort who often have only the most tenuous grasp of maths and physics.
So, yes, there are loose ends. But GR has loose ends as well. Where do we go next? Einsteins fields equations before first solo? Unified field theory before ATPL?
Physics is like a tool box. You use the right tool for the right job. In this case, newtonian physics is the right tool for analysing forces in climb, descent and turning.
Why do you want to invoke GR when by your own admission you have little understanding of it? (That's a rhetorical question by the way... it means don't).
Now, fascinating as its been, I am done with this thread. Its become very circular and tending to drift well off topic.
regards,
pb