Question: Why a simple theory of lift like the one described below, is not accepted?  

If you don’t agree with the following, please just skip this question and do not delete or downvote it, because it is a legitimate, scientifically sound explanation of lift.  

Common sense theory of wing lift fundamentals:  

Assume an airplane moves with the right speed and proper angle of attack.  

The wing pushes the air in the direct path of its angle of attack. After the wing passes, the air previously in front of the wing, ends up down below the trailing edge. Air when pushed compresses, thus creating a higher-than-ambient pressure. The higher pressure then propagates and dissipates with the speed of sound.  

The upper surface of the wing moves away from air, thus creating a lower-than-ambient pressure. Air from around then rushes in to fill the void (lower pressure), also with the speed of sound.  

The pressure difference translates in a force pushing the wing back up. That force is Lift plus Induced Drag.  

There is more to it of course, especially about what a proper angle of attack is, and what the right speed is.

Answer: 
Pressure is the result of momentum exchange between gas particles and a surface. Even in a gas which has no bulk velocity vector, each particle travels in a straight line until it collides with either a surface or another particles. When a particle collides with anything it can’t continue on its original trajectory therefore per rules of classical mechanics it must undergo an exchange of momentum to change its velocity vector. The rate of impulse of many of these particles is simply expressed as a pressure. This simplification is used since we are only interested in the net effect on the surface/body and not the individual particles. When the bulk / free stream gas has a net velocity vector the added kinetic energy of the particles will be accounted for when they collide with a surface. This results in an increase in the impulsive delivered by individual particles as well as a net increase in the total effect the gas has on the surface.

The lower pressure on top of the wing is due to the collisions on top of the wing not needing to redirect as much momentum as the bottom. In fact the bulk velocity vector of the fluid on top of the wing points away from the surface reducing the relative speed a gas particle will strike the surface with. For speeds much lower than Mack 1 compressibility is negligible due to the speed of the random motion of particles being much much higher than the speed of the bulk flow.
The main problem with that explanation is that it makes conflates lift with profile drag. Lift is a different phenomena and the "lift" in your explanation is less like actual lift and more like the upwards component of profile drag. The explanation contains just enough specifics to be wrong, or at least miss the primary point of discussion (what makes a wing, a wing what it is and not something else). That's why it doesn't work even as a simplified, qualitative explanation.

It would be sort of like saying both propellers and paddle wheels work the same way. Yes, they both push water backwards but they don't do it the same way. An early propeller like a screw propeller might produce thrust using using drag principles like a paddle wheel does, but proper propeller, as we know it, does not and instead operates produces thrust via lift principles. So by implicitly describing it to operate based on drag misses it for what it is.

The simplest explanation I have heard for lift which is vague enough to not be wrong, as far as I know, is that the wing "throws down air" to produce upward momentum but this leaves beginners to ask how the top surface throws down air at all since supposedly it plays an important role.

That leads to the second simplest explanation which is that due to the properties of fluids, the air follows the shape of the wing and is guided downwards which is something that cannot be reproduced with particles such as sand.