I think subtle expression possibilities is the key. Piano, electric piano and organs have a large and very finely controllable dynamic sound range1 (either by true continuous 𝆑𝆓𝆏 spectrum right "at the fingertips", or lots of of possible organ stop/drawbar combinations), so you can always counteract where it might get tiresome, without however necessarily creating an obvious and possibly distracting sound effect like the tweaking of a synthesiser's filters all too often constitutes.
Now, the question is what possibilities would a synth have to incorporate such subtle expression. The most obvious thing is to make some parameter[s] controllable particularly fine and intuitively – that's why almost all synths now have at least a big mod wheel today, usually in fact some joystick etc.. That works really well for melodic playing, but quickly runs into trouble when you try to serve more of a pianist role:
- Using your left hand to tweak such controls is a big problem when you need it to play accompaniment, especially if there's no bass (which you could play with your feet on organ pedals)
- Even when the parameter is steered by expression pedal or channel-aftertouch so you have both hands free to play notes, you'll only be able to add expression uniformly to all played notes. That makes it impossible to properly emphasise the "virtual voices" that are hidden in a well-arranged piano or organ score, short of selective dynamic emphasis or placement of voices on different manuals.
So that doesn't do the trick. Now, we could obviously copy the ways you control expression on the traditional instruments. All keyboards are velo-sensitive nowadays and can be used to control piano patches reasonably well. Great! Then, what do we route this to? Most obvious would be just VCA volume control, but that's hardly versatile enough. Filter cutoff and modulation / vibrato are good candidates (not without reason the typical mod-wheel parameters!) but here you'd really need continuous control, over the entire note duration. That's precisely what velo can not. Effectively, the only thing velo control does really well is a combination of volume and timbre – and then it'll invariably be a "piano-like" sound, at least roughly. With chorus-ey modulation it'll sound like DX-7 e-piano2, with short sustain like something between marimba and clavinet. At any rate not "typical synthetic".
For organ-like expression much the same issue: an additively composed sound will sound organ-like. Combining multiple manuals meaning you're effectively switching between patches, as you ruled out.
So, does that mean synths can't offer anything new when it comes to "consistent solo sounds"? I think they can! Parameters like filters and modulation, perhaps also intonation per-note can give amazing expressive possibilities. Only problem is, it used to be technically infeasible! Well, polyphonic aftertouch has long been doable, but the CS-80 wasn't exactly affordable. It's just really difficult to implement something like that with analogue technology.
This is the digital age however! The Haken Continuum has been around a while already – again, not quite cheap; also it's not really a keyboard-thing. But there are a whole lot of innovations in more-or-less–keyboard-like controllers. The Seabord adds some keyboard haptics to the continuum concept.
But my favourite candidate are Andrew McPherson's TouchKeys. You retain a true familiar keyboard, but with two dimensions of expression added for each note. (Saidly the first production was just the KickStarter-backed one; there's a waiting list for a second run right now.)
1Crucially: unlike when just turning some volume pot, pianos actually change the sound character a lot depending on how hard you strike the tones.
2Coming to think of it, the DX-7 may be the "historical synth" coming closest to what you seek.