Absolutely right about the racing car.
However, an airplane differs from the racing car by having a huge inverted spoiler forward or the aft spoilers, which is creating lift!
If a wing is capable of creating a 50 tons lift, after spoilers are extended and pitch is zero when on the runway, it will still generate, for example, 10 tons of lift.
Now, the tailplane is designed to create a constant negative lift.
By pushing the yoke fully forward, I think you can reduce this negative lift, or even create a positive one - it doesn't matter - by, say, 3 tons (imaginary figures).
Doing so your pitch will decrease by a couple of degrees.
This may reduce the residual positive 10 tons on the wing to 5 tons.
You see from all this simple math that the net result is in favor of pushing, because the apparent weight of the airplane has increased by 2 tons (-3 on the tail, +5 on the wings).
Thus more weight on the wheels.
Of course all this is type and conditions dependant.
If conditions permit a further compression of the nose oleo, for example, it's better to push, otherwise it's better to pull.
If the imaginary math above is in favor of pulling on a certain type, so be it, pull.
And the list is long.
That's why it's almost impossible to establish an official procedure.
Ps: an instructor once criticised me for pushing too much after touchdown.
He said Boeing's official technique was to neither push or pull.
Just to leave the yoke neutral.
I can't recall such a statement in any book. Anybody has got the official reference?
LEM