It's the other way round. The vectors-in-state-spaces representation is necessitated by the fact that it is a probabilistic theory, and has no physical content

It's the other way round. The vectors-in-state-spaces representation is necessitated by the fact that it is a probabilistic theory, and has no physical content itself besides that. The first half of "The Structure and Intepretation of Quantum Mechanics" by R.I.G. Hughes [1] lays this out very clearly. He derives most of the framework simply from the premise that it's a probabilistic physical theory. Then he shows how you can add a single extra element to it (eg. unitarity [2]) to turn it into full blown quantum mechanics (i.e. derive Schrodinger's equation), and lists several alternative ways you can do that. It's a great book, and accessible to the layman. [1] https://www.hup.harvard.edu/catalog.php?isbn=9780674843929 https://www.hup.harvard.edu/catalog.php?isbn=9780674843929 [2] https://en.wikipedia.org/wiki/Unitarity_(physics) https://en.wikipedia.org/wiki/Unitarity_(physics)