To summarise those desirable qualities, a good notation is:
Composable
Shareable
Degrabable
Learnable
Semantic
Grammatical
Efficiently expressive
Suggestive
Intentional
Legible
I can’t quite tell whether I’m trying to force together two concepts (notation and user interface) which are fundamentally different, or whether there is something fruitful in seeing them as aspects of the same thing.