Bernoulli works: it describes the pattern of pressures over a shape, and by integrating those over the entire shape, you can accurately predict the total forces and moments acting upon that shape.
Newton works: it describes that for every action there is an equal and opposite reaction. If you are able to measure or calculate the mass flow of air due to the passage forwards of the aeroplane, you'll find it corresponds exactly to the lift.
Now listen carefully: if you know the pressure distribution over the shape travelling through the air, and use this to calculate how much mass of air should be displaced downwards (using, as it happens, Newton's second law). Guess what, you get the right answer, which Newton's laws will then allow you to calculate total forces with.
If we really want to get complicated, we can come to the Navier-Stokes equations, which are in fact derived from a form of Bernoulli's equation (there are several - the incompressible, compressible, and unsteady forms - in ascending order of difficulty to understand). The N-S equations (together with some clever maths called "Cauchy's integral" and a general principle called "vorticity" are basically a way of taking the shape of something and (guess what) predicting the pressure distribution over a shape. From that, Bernoulli's equation(s) allow us to predict forces, and Newtons laws allow us to predict mass flow effects.
They really are all interlinked and inextricable ways of describing the same thing. It is totally incorrect to say that any portion of an aerodynamic force is due to any one, and not any other.
It is of-course true to say that any of these theories can be described also as simplistic - but rarely so simplistic as to be unuseable. Last week I was busy working out the equations for an obscure type of airspeed indicator - using the simplest incompressible form of Bernoulli's equation, and with accepable accuracy. (I'd rather have been flying, but that's another story!).
G