I agree with the reason for the force, I disagree that this cannot be characterised as weight.
As I said, the observer in the aircraft would have no way of distinguishing which part of the forces he was feeling were due to gravity, and which to acceleration, just as an astronaught in free-fall has no way of sensing whether he is in orbit or in deep space.
Do you agree that an Astronaght in orbit is "weightless" even though he is still subjected to a force very close to 9.8N due to gravity, but neutralised by his orbital acceleration?