The biggest piece of evidence that properties don't have set, distinct values prior to measurement is given by Bells Theorem and the experiments to test it. If you send an entangled pair of photons out, one west and one east, and measure their spins - and you do that thousands of times - then if you have a normal classical idea of causality, it becomes extremely difficult to square the gathered statistics with a classical world view.
These experiments don't entirely rule out all types of determinism, but they do rule out some specific classical views of determinism.
So it rules out some common conceptions of determinism, and it doesn't rule out any conceptions of randomness, which is why it might be seen as evidence for randomness.
In addition, keep in mind that quantum mechanics is taught, in many universities, almost by default like the Copenhagen Interpretation is the right way to think about it - Copenhagen explains Bells Theorem results with a combination of randomness and non-local wave function collapse. Since that's the most common, and pedagogically default, interpretation, it will naturally look like an experiment which proves Quantum Mechanics must be the case and Classical Mechanics cannot be is also giving more weight to randomness.