How much effect does the Bernoulli effect have on lift?

There's no problem with the Bernoulli effect, only with the way it's understood and explained. It's usually explained with mistakes, like the need for asymmetrical airfoil and equal flow time above and below, and without mentioning the need to deflect the direction of airflow.

Here's the best light-math explanation I've seen. Also study this section that directly answers your question.

EDIT: It is easy to find wrong pictures like this:

as opposed to a correct one like this (from the link above):

So the answer to your question is: All of the lift depends on the Bernoulli principle, because speed and pressure are in trade-off, but the physics need to be correctly understood.

Sometimes you will see statements like 'some of the lift is caused by Bernoulli's principle and some of it is caused by Newton's laws", but this is the wrong way to think about it. The fact is that 100% of the lift can be explained by Newton's laws and 100% can be explained by Bernoulli's equation. Both approaches explain 100% of the lift.

The problem is that popular explanations using Bernoulli's principle usually make a dog's breakfast out of things, so I understand why you think it's a "flawed" explanation - usually it is. However, applied correctly Bernoulli's equation can be used to predict 100% of the lift.

To grossly oversimplify things, classical aerodynamics calculates lift something like this:

• write down the differential equations expressing conservation of momentum, conservation of energy, and conservation of mass.
• solve those differential equations and apply boundary conditions to get a solution
• the solution is a vector field that represents the speed and direction of the airflow at each point in space
• Forget about the direction and just use the magnitude of the vector - this is the speed
• Now that you know the speed everywhere, use Bernoulli's equation to substitute pressure for speed
• integrate the pressure over the airfoil surface to get the lift.

Done correctly, this number is 100% of the lift. So, Bernoulli's principle is responsible for 100% of the lift. The thing is, almost all of the physics is in step one - the rest is a bunch of hairy math and then a calculational trick at the end to use Bernoulli's equation to turn speed into pressure. This trick (mis)leads some into thinking Bernoulli's equation somehow "explains" the physics, but without the context of the rest of steps it makes little sense.

If you look at the overall context you'll see that Bernoulli's equation is a small and relatively unimportant part of the overall theory. But correctly applied, it predicts 100% of the lift.