Gravity is not going to pull the nose down, the plane is flying based on a center of mass, the CG (center of gravity is very important in flying). The only time gravity pulled "my nose" down, is when I was going straight up (T-38) and hit zero knots, then the plane stopped going up, and the nose fell straight down, and we dropped until we could fly again in air. If the CG is off, the aircraft can crash, see the cargo jet with the load shift, it crashed, it failed to fly after a massive CG shift to the rear. We load aircraft to maintain a safe CG. Flying is a balancing act, we don't care if the earth is round, we are flying in air on the earth. I never trimmed the KC-135 to adjust for the curved earth on 9 to 10 hour flights from Okinawa to Diego Garcia.I think this may be a point where people are dismissing the FE claim too easily. A car will keep it's nose on the ground as the earth curves because of gravity, but there doesn't seem to be any reason why gravity would pull an aircraft's nose down in the same way.
Let's say I design an aircraft that maintains exactly level flight in a wind tunnel at a specific air pressure and air speed. I lock the control surfaces in place and then attempt to fly that aircraft across the Earth at the corresponding pressure and speed (ignoring all the obvious complications like variations in wind, pressure, temperature etc). It seems as if some people are claiming that this craft would maintain level flight in this case as well, "automatically" pitching itself to follow the curve of the Earth. I don't see why that would be the case, as gravity would be pulling on all parts equally, not the nose in particular. So it should indeed seem to slowly pitch up, gaining altitude until it eventually stalls.
In that sense aircraft do compensate for the Earth's curvature, in that they must be trimmed to pitch ever so slightly down compared to what they would on a theoretical flat plane or in a wind tunnel in order to maintain a constant altitude. As has been said already, that compensation is so miniscule as to be lost as noise amongst all the other more significant factors, but it is still there. So, I think when people ask "why don't planes have to constantly pitch their nose down to account for curvature?" the answer should really be "they do", rather than just dismissing the idea as silly.
The plane is flying in air, there is no need to pitch for the curvature of the earth. We trim the plane for level flight and make adjustments as we go to compensate for weight change, CG change, and when we are not free to fly a set AOA (like max Range AOA), we must constantly make small changes to hold an exact altitude per standards. If we have an autopilot the altitude is automatically maintained. The Autopilot was designed by assuming the earth is flat and not spinning, because the math is more complicated for a spherical spinning earth. For an aircraft flight dynamic the effects of a spherical spinning earth are negligible (but not for navigation, we must model spherical spinning earth).
We can assume a flat earth for flying. Designing a plane in a perfect flat earth wind tunnel to fly level, it would fly level. The autopilot which flies commercial aircraft already assume a flat earth, and it works. The only pitch changes are made to maintain a pressure altitude, changes in CG, changes in weight. If I fly by hand and have the plane perfectly trimmed, I add thrust, we go up, reduce throttle, we go down, it is that simple. For subsonic flight and maybe up to and past MACH 3, we can assume a flat earth, there are no changes made for the curvature of the earth, or coriolis. If we do the math based on a curved spinning earth we find the terms related to curve and spinning are negligible, we then make the assumption of flat earth, no spin for the flight dynamics, the flying part.
If it helps, think of a particular pressure altitude, FL350, as an ocean of air, and we are at that exact altitude, which does automatically curve with the earth, no need to pitch down each mile eight inches, we are already there as we fly. I prefer to fly max range AoA, or use 99 percent max range on the high side, which I usually used .81 MACH in the KC-135 on the MACH indicator after adjusting for errors in the system.
The aircraft has no clue the earth is round, and flies steady at an altitude when trimmed properly. As we burn fuel the lighter plane will rise or go faster, based on holding an altitude or free to roam in altitude.
When developing flight systems assumptions are made, and we can assume flat earth without a problem. Irony, the plane you designed to fly on a flat earth perfectly trimmed is how we do it, and gravity acts on the entire plane, not the nose, and at each instant we are level there is no need to pitch for a curve, it does not make sense for flying.
This makes it easier to do the math, and it worked, because this is how our aircraft and their systems for flight are designed. Plus, there is no doubt the earth is a spherical spinning mass.
Let's take an aircraft trimmed for climb, throttles set for climb, what happens. Payne Stewart, his flight crew failed to pressurize the aircraft properly and the crew and passengers passed out because they failed to recognize their hypoxia signs. The aircraft was not being flown by anyone, and continued to climb set trim, no changes, and did not stall, but reached a service ceiling for the specific throttle setting. The aircraft ran out of fuel and descended and crashed. The plane automatically followed the curve earth.
Accident report for Payne Stewart's aircraft, trimmed for climb attained max altitude with a set pitch/trim the entire flight. An altitude based on a set pitch/trim, set throttle, no adjusting for the curve of the earth. An aircraft has to maintain a pressure altitude usually, the trim changes to account for weight changes, CG changes, etc. With set throttles to maintain an altitude, as the weight is reduced/fuel burned, the plane goes faster, the pitch is changed to maintain an altitude. Usually an altitude is set, speed control by reducing throttles as we burn fuel. No pitch change for the round earth. If we gain lift flying east, the pitch would still be set, not changing due to the curve, and if flying west, it would be set. The effect of round earth would be related to lift, no need to pitch down to follow a curve. It makes no sense.
Payne Stewart's accident report.
The accident reports for Payne Stewart's aircraft, after cleared to FL390 the crew passed out and left the plane in a set trim, as you suggested, no change in trim/pitch. It automatically follow the earth's curve. I can trim an aircraft for level flight at a throttle setting, if I increase the throttle, we climb, if I reduce power, we descend. If I level off and hold altitude and increase the throttle we go faster at that altitude. etc
There is no pitch down, anything related to the round earth would be a lift component, it is negilagable for most aircraft below MACH 3. , climbing, and descending flight.
Aircraft don't pitch down, aircraft follow the earth at a pressure altitude, there is no pitch down, it would change the speed of the aircraft to "pitch" down, and you would not longer be at the right pressure altitude. A flat earth would not have anything like we experience. As you walk away from the center of the flat earth you would have to lean, and flying would not work the thickest atmosphere were be stuck in the center of the flat earth, with almost no air at the big wall of ice, and the water would all be at the center of the flat earth, or in silly reservoirs on hills, where the water would not be level. Thus flat earth has no gravity, but some crazy claim of density which has people and things accelerating toward the flat earth, no gravity. What is acceleration on a flat earth. Without gravity there is no air on earth, no flying, no joy. The OP level experiment, the person doing the experiment denies gravity exists.
If we pitch down, we would go faster and faster, by the time we get to 6 degree pitch down, the aircraft would exceed the max speed and crash at high speed. At each instant we are flying at a set pressure altitude, we don't change the pitch or trim for a round earth as we fly, we remain in level flight.
To understand pitch and power, take a look at this.
Pitch for airspeed, power for glideslope. Lower pitch go faster, higher pitch go slower, power to go up, reduce power to go down. It makes no sense to dip the nose to follow the curve of the earth. Dip the nose, pitch down, you go faster. Any lift associated with a curved earth is part of flying, and you would trim your plane for level flight, and in a perfect world, no weight change, no CG change, no pressure changes, no need to change anything once level with set power. Payne Stewart's plane was at a set trim/pitch, and it crashed when it ran out of fuel, and climbed higher as it lost weight, and most like.y oscillated in altitude, because things change. The level experiment is not evidence for a flat earth. A flat earth would be evidence... and gravity does exist.