In conventional treatment of QM, one assumes that (1) physical states are normalized vectors in (rigged) Hilbert spaces and (2) operators correspond to observables, with their eigenvectors denoting states where the observable is equal to the corresponding eigenvalue.

From this, almost everything else follows, if we don't aspire to a von Neumann level of rigour. In particular, (1) is enough to derive that the evolution operator $U(t)$ is unitary and thus the infinitesimal evolution operator $-iH$ is anti-Hermitian, and (2) in conjunction with $\mathbf R$-valued observables in real life means that the observables are represented by Hermitian operators.

A better approach is to recognize that the normalization in (1) means we are now talking about projective Hilbert spaces. (2) then transforms into expressions like $\mathop{\mathrm {Tr}} \rho A$, $\rho = \psi\otimes\psi^\dagger$, etc. The general framework is clear to me. However, I’ve yet to see a text that tells this story systematically from the start, without references[1] to the conventional viewpoint, and in particular, points out the exact set of things we must take on faith to proceed in this manner, like my sketch above. Can anyone suggest such a text? A textbook would be perfect, but anything below the level of Deligne would do.

[1] What is called normative references in engineering standards, not comparisons, of course.