I thought what was the major reason for pushing the stick while braking has been almost totally overlooked: to decrease the residual lift from the wings, thus getting MORE weight on the main gear.
In simple words: while it's true that UP elevator will tend to increase somewhat the apparent weight for reasons discussed at length here, it will also increase the incidence of the wing, while elevator full down will reduce the attitude by maybe a couple of degrees.
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Elevator full down (i.e. stick full forward) will only effectively weight transfer onto the nosewheel, leading to "wheel-barrowing" (a condition of directional instability). It won't, by any useful measure, decrease the wing's AoA. Why? Think of a depressed oleo as only being able to soak up shock i.e. weight-bearing oleos are effectively incompressible to flight-control inputs - therefore fully-down elevator will just load up the nosewheel. It won't depress the nose oleo appreciably more, nor take any angle-of-attack off the wings. By contrast backstick will load up the mainwheels, increasing traction, enhancing directional stability and you will achieve effective braking much earlier - particularly on a wet or contaminated runway. Why wouldn't backstick raise the nose? Don't forget the combined pitchdown effect on nose touchdown of engine reverse, braking and spoilers. During the important period for effective braking this pitchdown couple enables the backstick's effect of loading up the mainwheels - by stopping the progressive up elevator from raising the nose. Why progressive? Well obviously a pilot is not going to immediately put his yoke to the backstops after nosewheel on. But as speed decreases, inevitably, if going for the maximum backstick braking effect, the yoke
will end up fully back. Obviously differently configured airplanes will have slightly different characteristics.
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...by pushing the stick (almost) fully forward, as I've done since now, what do we get out of the balance?
More lift from the tail, or more-reduced lift from the wings?
You will note that internationally well-known Experimental Test-pilot John Farley came out in support of the backstick braking technique but was rudely rebuffed, essentially in mid-post, by one of those who cannot accept the practical facts..... and who instead waxed on with ever-confusing hypotheticals. Backstick braking is a proven effective stopping technique. Unfortunately it's not yet been automated and, because it is a dynamic process, it is easily misunderstood. Those who normally oppose change have been well represented on this thread and have used quite illogical reasoning in an attempt to deny its effectiveness and conjure up fanciful possible dangers.
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This lift reduction on the wings by a pitch decrease is the key to this issue, and I'm really surprised nobody has given this the due attention.
Or have I missed something?
You most certainly have missed the big points essential for understanding how the technique works.