Bernoulli applies and Newton applies. Can't have one without the other.
However: One thing which is absolutely and provably untrue is the equal transit time theory often taught as the basis of the Bernoulli explanation.
Air parcels separated at the leading edge do not meet up again at the trailing edge. There are plenty of pulsed smoke wind tunnel and CFD imagery available to show this. The air going over the top surface of the wing gets to the trailing edge first.
Regarding the pushing and pulling, they are both the same. The bottom of the wing pushes air down, which goes hand in hand with a relatively higher pressure. The top of the wing sucks air down, which goes hand in hand with a relatively lower pressure. Whatever velocity change you put air through comes with a pressure gradient to match.
To get into a discussion about which surface generates the most lift is opening a can of worms. At low angles of attack, the air pressure at the bottom surface of your typical wing will likely be covered by a pressure lower than ambient, which in fact means it is generating sink rather than lift. How much sink is equal to 1/3 of the lift generated?