The air isn't as such stuck to the moving ground.
the confusion here is because you making it very complicated.
The propellers job is to transfer power to the air which results in a force towards the rear of the aircraft. The air is not attached to the belt so it is free to be accelerated by the prop producing the force.
The wheels on the ground are just there to stop the prop hitting the ground.
While taking off they have minimal resistance to forward motion but in the engineers brain all it is, is a force resisting the plane going forward in the oppersite direction to the force produced by the prop.
Now the wheel/ belt is treated as a system in it self. The sum total of all the proccesses going on will produce a resisting force. And usually in the grand scale of things it will be forgotten about by engineers because its less than 1% of all the other forces.
Now you add all the forces up resisting the forward motion of the aircraft so thats the forces due to the wheels, forces due to aero drag and you subtract them from the force produced by the propeller. If the resultant force is positive the plane will accelerate forward relative to the airflow if it is negative it will go backwards if it is zero it will stay stationary.
Its a bit like all the stuff to do with the difference between Ground speed and airspeed when your doing nav.
With any system engineers are taught to simplify first down to just the basic forces involved and not really care how they are produced. Then form whats called a free body diagram which graphically shows you whats going on. Some get by with just sticking numbers next to the arrows and some preffer to draw the arrows to scale. In this case you would have. Where x is the plane.
T<---------------------------------X->D
Now the problem is that experence allows the engineer to forget about some things because they know that it won't matter for that problem. We tend to work to 3 significant figures any thing less than that just gets forgotten about because of all the other presumtions and arse covering with materials means you will be very lucky to get an answer that will be correct to anything more than that anyway. You can see from the diagram that the force forward is way more than the force the other way. This would be at the start while the aircraft isn't moving relative to the air flow.
As it speeds up the aero drag increases so it becomes.
T<---------------------------------X------------->D
Then once it becomes airborne the drag produced by the wheels is removed. Thats why some times you feel a little bust of acceleration as the wheels lift.
And the plane continues to accelerate until.
T<---------------------------------X--------------------------------->D
Then the plane is at a constant speed relative to the airflow.
So whats happening at the ground is really irrelavent to whats happening to the airflow around the wings and prop. The wheels ground produce a force and the propeller produces a force. If the force from the propeller is greater than the force from the drag from the ground it will move forward. Which will then make airflow over the wings which will produce lift for flight.
MJ