You have mixed up some different ideas. The atmosphere generally rotates with the Earth due to friction and gravity effects, but there are complications which caused you to reject the whole idea, whereas you only need a more nuanced understanding.
For an aircraft in the air, to the first order the plane and air are both moving with the surface of the Earth at no relative speed. But the direction of motion is changing all the time since the path of a point on the surface is circular (once round per day). SO as the plane approaches the runway, the runway is rotating at the same rate as the Earth, at 360 degrees per day or one degree every 4 minutes. The plane and runway are both moving at the same speed due to the Earth's rotation, but as the plane is not fixed to the solid surface the air must supply the forces to change the heading of the plane. In amongst all the air buffeting that a plane must correct for, the slow rotation will not be noticed.
The second effect is the Coriolis effect, which is real and observed to cause cyclonic winds. If you are in the UK (say, N 53 deg), the air is moving East with the Earth at about 1000 km/h. If the air moves North a little, the Earth there is moving at the same rate of angular rotation, but because it is closer to the axis it has a lower speed (until the N Pole where the speed is zero). For a 1 degree latitude change North to Scotland, the rotation speed of the Earth is now 980 km/h, but the air still has its 1000 km/h Eastwards momentum, a difference of 20 km/h to the East compared to the ground. It has sped up relative to the ground simply by moving north, as if a force had acted on it. This is the Coriolis effect, and it applies to anything moving that changes latitude, from air movements to naval shelling.
You ought to take a look at the Foucault Pendulum experiments that
directly measure the rotation of the Earth to match that seen by observing the stars.
https://en.wikipedia.org/wiki/Foucault_pendulum