I consider Bernoulli, who explains how the pressure differential is created, to be a subset of Newton, who explains why the pressure does anything.
The "feud" between advocates of only one or the other is just silly!
Looking at the edges of the original square plate, that's a lot of air bending those edges down.
But, it's like blowing the fan into the sail. If the fan is attached to the boat, the boat can't move.
And the heli can't lift.