I am not a Flat Earther. I understand that the plane's trim, speed and atmospheric pressure determine the plane's altitude. As the plane moves forward the Earth does indeed curve out from under it and the plane does indeed try to fly off into outer space and fails to do so as the air pressure becomes too thin for the trim and speed and so the plane constantly settles back to the correct altitude, the altitude that has enough atmospheric density for the trim and speed to stop the downward settling. Thus we have level flight. Just one problem, even though the plane stays at one altitude, the nose seems as if it should gain small upward tilts that slowly accumulate into a serious problem. How is the nose, not the plane, following the curvature of the Earth?
Good question! First practical detail to note would be that I doubt there is any airplane with such stability (without electronic aids etc) that you could left flying hands-off for long enough period to perceive effects of Earth curvature without other disturbances affecting flight path and attitude much more.
But lets take this as a theoretical problem: absolutely calm air, no disturbances, only curved Earth. The fact is that air density or pressure change is not the reason for curved path around the Earth. It would work even with constant air density (or pressure) all the way up. It is on the contrary the "automatic" pitch adjustment which goes first and as the pitch adjusts, flightpath follows it round around the Earth.
It is important to remember that actual pitch (or angle of attack) is result of balance of acting forces (gravity, lift of the wings and tailplane), it is in no way "conserved" due to inertia or whatever.
All starts with the mechanism how the pitch-wise stability of airplane works. For really nice and detailed explanation see "See How It Flies" by John S. Denker (whole text is only to recommend reading for more technical understanding of flight). The trim mechanism (balance between tail and wing forces) works such that it rotates airframe towards one fixed value of angle of attack and keeps it balanced there.
So what happens ... let's do the differentiation of motion in small steps:
- the plane flew perfectly straight a short bit in tangential direction to the surface;
- so the direction of the gravity force has changed by an infinitesimal angle and has a component pointing backwards relative to the airplane now;
- this new component of gravity force starts slowing down the flight (we have not adjusted any control or engine power);
- slower flight means less lift, so the airplane starts to fall down a little;
- falling down means that relative airflow will come more from underside increasing the angle of attack perceived by airfoils;
- now the trim mechanism comes into the play correcting the angle of attack to its original value.
So this small amount of falling down is corrected to forward flight (with respect to the airplane), but now the trajectory follows Earth curvature. Flying the circle means, that you are all the time falling down as seen relative to the previous direction of motion, so it is exactly what you need.
I‘ll try a very short answer:
While the physics are not exactly the same, it’s similar to why the noses of ships or cars don’t point up after travelling a while - the equilibrium state (trimmed level flight for an aircraft, trimmed level float for a ship and supported by all wheels for a car) follows the surface of the earth so as to be locally horizontal at each point.