No one integrates momentum flow, because it's basically computationally impossible, even for low Reynolds number flow.
Thats certainly not true (the part about computational impossibility). While I don't have much experience with fluid flow in planes, I have lots of experience with numerical fluid flow in GR (I like to say I spent my masters dropping stars into black holes). We integrate momentum flows across boundaries all the time to verify conservation laws and validate simulations. If you have the flow, its not computationally complex at all.
Not at all. But you can't compute it with a simple momentum-based "air goes down" argument, because you cease the ability to track the momentum of a fluid element once it leaves your cube.
Why would you ever need to track the fluid element once it leaves your cube? I think you seem to have some sort of idea that you need to keep track of things away from the plane, which is wrong. The fluid is only capable of exchanging momentum with the plane when it is near the plane.
Lets back off from order of magnitude for a second, and just consider the following- if we draw a large cube around the plane, integrating just the pressure won't be enough to give us the forces on the plane? We also have to take into account fluid flow at the boundaries?