accdd said:
if I repeat a measurement of the same observable in succession I get the same quantum state
You only get the same result and the same state if you repeat
precisely the same measurement. Put another way, how close the new measurement is to the old measurement determines how likely it is that you will get the same result. If there's an evolution between the two measurements that modifies the state so that the result of the same measurement would be different, that will be equivalent to there being no such evolution but with the subsequent measurement being different: this is directly comparable to the difference between the Schrödinger picture and the Heisenberg picture of states and measurements.
Sequential measurement is a thing in the literature. My recent article in JPhysA 2022, "The collapse of a quantum state as a joint probability construction",
https://doi.org/10.1088/1751-8121/ac6f2f, on arXiv as
https://arxiv.org/abs/2101.10931, discusses sequential measurements as part of a wider discussion.
Your one observable case can be thought of as a special case of a commutative algebra of observables. There is a point of view in which a commutative algebra of observables can be thought of as in some sense "classical", however if it were that simple we would not still be talking about interpretations of QM. You could try a paper by
Tsang&Caves in Phys. Rev. X 2012, Ref [17] in the article I just mentioned, "Evading quantum mechanics: engineering a classical subsystem within a quantum environment", which develops the idea of Quantum Non-Demolition meeasurement from inside QM, however there are
many approaches that come under the general heading "
Modal Interpretations" that develop similar ideas.