Somethere you said that you have no idea how to fix this.
Look how implemented same aspect in do a barrel roll mod. https://www.curseforge.com/minecraft/mc-mods/do-a-barrel-roll
It rotates player eye same way but rotation works absolutely correct.

Awesome, I'll check it out.

Also an idea: add the airplane management mode using the mouse and AD keys. By analogy with this mod. It would be very convenient.

Due to the fact that the mod has a GNU General Public License v3.0, you just can copy their code, slightly tweaking the coefficients. This would fix all the bugs related to the incorrect processing of the movements described by me in the previous issues. The mod has an understatingly more realistic and less buggy airplane engine.

From a quick look it seems like the mod is working with player rotation values directly, while planes use quaternions internally and only apply them for rendering the player. It would require quite a bit of rewriting, which I might try in the future.