To use it strictly in a 'traditional' sence, it'd be ideal to not be switching around between screens, modes, instruments, etc.
One would either have the Scales screen up, for fast switching between root notes (even in chromatic mode, switching can still have advantages), much like an Autoharp
Or that screen would display macros to control an instrument Dedicated and refined to be played especially via Push's features.
The things that make a classical instrument so playable are, a shape which demands experience using its layout, a tone that is expected given various nuances of play, and a collection of musical pieces which are written to be recited via that instrument.
These aren't absolutely necessary for brilliant playability, but of you want to have the device act as a serious instrument, those may be things to consider.
I'd also consider an extra side-controller of faders and knobs
