But once you decide you're going to interpret the wave function as distributing probability among some set of orthogonal subspaces, you're already compelled into the Born probabilities.
All you need to decide that you ought to do that is the general conclusion that the wavefunction represents some kind of reality-fluid. Deciding that the nature of this reality fluid is to be made of states far more specific than any entity within quantum mechanics comes rather out of the blue.
But the phrase "reality fluid" is just a place-holder. It's a black box labeled "whatever solves this here problem". What we see is something particle-like, and it's the dynamics relating our observations over time that complicates the story. As Schrödinger put it:
...[T]he emerging particle is described … as a spherical wave … that impinges continuously on a surrounding luminescent screen over its full expanse. The screen however does not show a more or less constant uniform surface glow, but rather lights up at one instant at one spot[.
If it's worth saying, but not worth its own post (even in Discussion), then it goes here.