The way I view the problem presently is that, indeed, quantum theory is a theory of contextual probabilities. This much we agree on: within each context, quantum probabilities are nothing more than standard Kolmogorovian probabilities. But the contexts are set by the structure of the Positive Operator-Valued Measures: one experimental context, one POVM. The glue that pastes the POVMs together into a unified Hilbert space is Gleason’s “noncontextuality assumption”: where two POVMs overlap, the probability assignments for those outcomes must not depend upon the context. Putting those two ideas together, one derives the structure of the quantum state. The quantum state (uniquely) specifies a compendium of probabilities, one for each context. And thus there are transformation rules for deriving probabilities in one context from another. This has the flavor of your program. But getting to that starting point from more general considerations—as you would like to do (I think)—is the challenge I haven’t yet seen fulfilled.
