So the mystery is the opposite: given that a mechanical understanding of the piano suggests that the pianist has no control of note timbre, why does it appear that they are able to adjust note timbre? I have my own speculations about this question, but it is not taken up by the article [1] which in fact only speaks of 'perceived timbre.'
In my own opinion, for the reason already given, there is no magic by which timbre is controlled through pressing the key daintily or pressing it with gusto, given that identical volume is achieved in each case. However the sonic environment is very complex and the pianist has considerable control over it through the dampers' connection to the keyboard, independent from the pedals. The process of listening is also very complex. The sense experience of hearing a staccato note is not the same as the sense experience of hearing the first 0.05 seconds of a note that later turns out to be sustained, so even if in fact the state of the piano is identical between the two cases within those 0.05 seconds I may perceive the timbres as different as my brain integrates the whole staccato note into 'one sound' while the sustained note is received as a continuous tone diminishing in intensity.
Ultimately I find the argument that they do not control timbre convincing, and that the piano (sans pedals) allows for as much control over timbre as the harpsichord does over dynamic. But as it is a very expressive instrument there is an illusion that is present for performance that cannot be reproduced through isolated notes. It's not a bassoon.
In that case, we can say that timbre of a piano can definitely be controlled separately from volume—as in, one can play a note with the same volume (to some precision), but different timbre—if nothing else then because a piano note does not really exist standalone, and the perceived character of a sound from a given key will always depend on what keys were played before and after, how they were played, etc., adding innumerable complex interplays between all the factors involved, even possibly including the movement of the pianist.
Describing this difference as an illusion may be meaningful in some contexts, I suppose, but they would be pretty narrow and it is not too different from classifying consciousness as illusion: perhaps it cannot be quantifiable in certain ways, yet in other (very concrete and important) ways it is more consequential than many things we can quantify.
In other words, so if you compare A then B then C vs. Z then Y then C (where B and Y could even be the same note, just played differently), you can have “the same” C as measured in separation from the rest[0], but which crucially would nevertheless have a different character in listener’s mind—which may well be subjective, but also practically most relevant, and evidently reproducible.
[0] If that is even possible, considering the tails of preceding notes would be audible.
There probably will be. We're used to thinking of key velocity as a constant, but the escapement mechanism is quite complex and isn't truly instantaneous. There's almost certainly some room for dynamic velocity variation in touch as the key goes down.
Likewise with short vs long notes. I suspect there's some unconscious variation in key release dynamics, which will have a subtle effect.
What you're talking about is the more complex musical effect of many keys at once, with complex variations in timing (intentional and unintentional slurring of chords), dynamics (intentionally highlighting some chord notes over others), and all of the resonant interactions between open strings, dampers, and the frame.
That would be much harder to analyse.
Absolutely. Most of my comment is arguing how it would be not really useful, though, because perceived timbre (a.k.a. the kind that matters) is by definition wholly subjective, influenced by surrounding notes, etc.