Bernoulli and Newton - Cause and Effect
Both the Newton (laws of motion, flat plate) and Bernoulli camps are more or less correct. Lift is created by accelerating mass downwards. In steady flight, the force created by accelerating mass (free air, engine exhasust) downwards is equal to the force on the aircraft due to gravitational attraction (ignoring minor effects such as buoyancy). The downwards deflection of the air is produced by the difference in pressure between the upper and lower surfaces of the wing predicted by Bernoulli's Theorem, although the mathematics describing the flow and its pressure differences is rather complicated.
(If we think of the example of a hosepipe playing on a flat plate, the plate slows down the water, so the pressure on the plate increases relative to that of the flowing water - which is the same as that of the surrounding still air, so the plate tends to move in the direction of the jet.)
Fifty years ago I was in the middle of a degree course in Difficult Sums which included, among many other things, a series of lectures on fluid dynamics. The years have eroded much of the detail from my memory, but the basic principles remain.
The partial differential equations that describe the steady three-dimensional flow of a real fluid are complicated, non-linear and do not have neat analytic solutions. When acceleration of surfaces in contact with the fluid is taken into consideration, the situation is even worse.
Without the assistance of modern high-speed computers, the accurate numerical solutions of the full equations for even simple cases are just not possible. The only feasible approach is to make simplifying assumptions, work out the numbers, and then perform experiments (for example with wind or water tunnels) to determine how closely the simplified theory matches reality.
The main sources of intractibility in the full equations are viscosity (or internal friction) and compressibility. So the simplest approach is to eliminate both, introducing the concept of the ideal fluid.
Solution of the simplified equations for two-dimensional flow round a circular object (equivalent to flow in three dimensions round an infinte circular cylinder) is (or was for me in the days of my youth) fairly easy. A simple trick (called conformal mapping) can transform this flow into the flow round an aerofoil shape. The difficult bit is finding a transformation that converts the circle into the aerofoil of interest.
This simplified model works well for some things. It predicts flows (and thus pressures) fairly well, so long as you don't look too close to the trailing edge of the wing. The pressures over most of the upper and lower surfaces tally well with the predictions from Bernoulli's equation. However this simplifies theory has one major drawback - it predicts zero lift (and drag). Introducing "circulation" into the theoretical flow can get out of this problem and produce acceptable predictions for many purposes.
However, the simplified system is not good for predicting flows near the trailing edge of the wing, or where the flow will separate from the wing with the resulting turbulence and increased drag. To model such things as flow separation and the onset of turbulence, the equations have to take account of viscosity, which increases the complexity of the computation dramatically. One of the complications is that there is a strong interaction between viscosity and typical lengths of objects in the flow. In one of the possible simplifications for viscous flow, it turns out that there is a characteristic number, the Reynolds Number, that identifies flows that are very similar though with widely different lengths and viscosities.
By varying viscosity as well as the scale of a model it is possible to set up an experiment in a relatively small space that can provide information about real flows past larger objects in fluids with different viscosity.
Reynolds Number modelling is applicable only where the fluid can be trreated as incompressible. It is initially a little surprising that air can be treated as incompressible for aerodynamic purposes up to speeds fairly close to the speed of sound. Beyond this, into the transsonic range and beyond, a different simplification of the general fluid flow equations is needed.
At subsonic speeds, the air can effectively "hear an aircraft coming" and move out of the way, resulting in relatively simple flow round the aircraft. At supersonic speeds, the aircraft arrives without warning and the result is a shockwave. The flow regime is quite different, and a wing designed for supersonic speeds is likely to be quite thin and symmetric, contrasting to the thicker and cambered low-speed wing.
Computational Fluid Dynamics (CFD) software typically will use very general forms of the fluid dynamics equations, modelling both compressibility and viscosity. Some simplifications may still be used, but fast computers permit much more detailed computations in a given time than are possibly with simpler tools.
So, rougly speaking, Bernuolli explains the cause of lift (the pressure difference), Newton explains the effect of the pressure difference.