Timbre is what makes the violin sound different from the bassoon, what makes the strings in general gives them their sound, and brass theirs. Timbre is what makes a singers voice unique.
Classical composers had a set of fixed timbres to play with, each instrument having their own. With synthesizers everything is possible.
There are two levels to "everything is possible." First, recorded music is a string of numbers, and any way of generating those numbers will sound like it's intended to. But synthesizing a desired sound is still an evolving process of developing useful interfaces and waveform generators.
Generating that sound in a live performance that is partially or fully improvised is a harder problem.
The evolution of solutions to either problem depends on which instrument you're interested in, and is based partially on Means, Motive, and Opportunity. There's a reason to synthesize a piano or Hammond organ, which is that they're hard to transport and maintain. Less motive for synthesizing a violin.
"everything is possible" except for a synthesized instrument that actually has the expressive range of a real violin or saxophone. Even a synthesized piano is noticeably lifeless compared to the real thing (synthesized piano bass notes are especially thin).
It depends on what synth you're using, and how you're controlling it. Most synthesizers and controllers are profoundly lacking in expressive control, and of course a synthesizer that just plays back a recorded sample of a real piano isn't going to sound as good as being in a room with the real thing. Just because the mainstream synthesizer world has been sort of stuck in a rut for 40 years doesn't mean this is the best we can do.
There is, however, some real progress being made. For controllers, check out the Linnstrument, the Osmose, the Continuum. MPE was added to the MIDI spec a few years back, which makes it a lot more feasible to sell expressive instruments and have them "just work" with existing synths. As for synthesizers, just about anything is possible. Physical modelling is becoming more popular.
A real piano is also profoundly lacking in expressive control over timbre (excluding extended techniques like half-pedaling or directly touching the strings that hardly ever show up in the repertoire). The only control the piano gives you over note timbre is how hard you press the key. Once you've done that, the hammer disconnects from the key and flies through the air out of your control.
A keyboard with polyphonic aftertouch, such as the Yamaha CS-80 (famously played by Vangelis), is objectively more expressive than a piano.
> The only control the piano gives you over note timbre is how hard you press the key.
This is just not true at all. The action of a piano is a very sophisticated mechanism that conveys both acceleration and velocity, along with the initial position of the key (e.g. fully released or half-pressed). This allows for a huge amount of tonal variation, though it's more subtle than a ribbon controller, and obviously a synth can do many things than a piano cannot.
All the complexity you mention only affects how the repetition lever (part 9) briefly bumps against the hammer shank (part 8). The only information this imparts to the hammer is velocity, and once it happens there's no more contact until the hammer falls back down. The hammer can't somehow remember what caused it to travel at a specific velocity and strike the string differently. There's only a single degree of freedom in its movement. What you perceive as "a huge amount of tonal variation" is mostly note timing.
It's entirely true for the attack, because of the escapement mechanism. At the point at which the hammer makes contact with the string, the hammer is mechanically disconnected from the key and moving purely through inertia. Once you've tripped the escapement, the hammer is beyond your control and pure physics takes over; the only variable is the velocity of the hammer when the escapement is tripped.
The release phase is controlled by a single parameter, albeit one that can vary over time. The key is holding up the damper, which returns to the string under gravity. A continuous value representing the range of motion of the key would be sufficient to model this with absolute accuracy. Digital pianos invariably sense release velocity rather than key position, because it's much cheaper to implement and is perfectly sufficient in virtually all cases; the range of motion between partial and full damping is so limited as to provide only a very limited degree of expressive control.
I don't have a good analytical response to this. Your comment doesn't pass the smell test for the following reason: I am a fairly serious jazz guitarist, and it seems to me like most great guitarists can be quickly identified by ear after playing three notes: Charlie Christian, Joe Pass, Blind Willie Johnson, Mary Halvorson, Django Reinhardt, etc. In some cases just one note is enough: B.B. King, Wes Montgomery, T-Bone Walker, Jimi Hendrix. (60 years later and still nobody plays octaves as well as Wes did).
Obviously some of their sound is tied to specific guitar models, but their tone is still easily identified on other guitars. And although these players were creative improvisers, that's not what I mean. Simply put: if it's a great guitarist, you can hear their hands like you can hear a singer's mouth. (Though hands are clumsy compared to mouths, and guitars are clumsy compared to horns.)
I am not anti-synth or anti-electronic music. But I do not believe any existing synth controller provides the expressive possibilities of an electric guitar, even Allan Holdsworth's goofy breath controller. Maybe I'm wrong and there's a great example out there on YouTube. But it seems to me that for an individual to express their individuality with a synth, they need to play a longer musical fragment and color their expression with timing, volume, vibrato, etc. Three notes "with pizazz" will not really be enough.
There are a lot of famous guitar players, but there really aren't a lot of expressive controller players. I don't know of anyone who is famous for playing the Linnstrument, or the Osmose, or Continuum.
Vangelis and the CS-80 is maybe an exception, though even if his songs are very recognizable, probably a lot of people could convincingly imitate him.
Carlos was great at deliberate performances, but if there are any 'Carlos Live' recordings, I haven't heard one! But then LOTS of people grew up learning to play instruments from great teachers for centuries ... not so many with synths.
Only the the best grand pianos have any bass. Your cheap 43" tall console piano for apartment use will not have much bass to speak off. The strings are too short, the box is too small. We have to restrict ourselves to True Scotsman's pianos.
Bass is easy to obtain out of a synthesizer: just make sure you have a subwoofer in the speaker system. It doesn't have to be a True Scotsman's synth or subwoofer.
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Classical composers had a set of fixed timbres to play with, each instrument having their own. With synthesizers everything is possible.