It seems like some experiments on quantum systems, like the electron $g-2$ measurement, do not rely directly on the Born rule, since they are more so measuring inherent characteristics of the evolution of the wavefunction.
Whereas, an experiment that obviously does rely on the Born rule is the single-electron double-slit experiment, since in principle you could tell whether the distribution of electrons looks more like $|\psi|^2$ versus, let's say, just $|\psi|$. However, I've watched a couple videos of that experiment, and to me it doesn't look precise enough to distinguish between $|\psi|^2$ and $|\psi|$. (I mean, it's a mammoth achievement just to get that experiment working at all)
So maybe it's just my utter lack of experimental knowledge, but I'm wondering, can we really be sure the probability goes as $|\psi|^2$ and not $|\psi|$?