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Before the late 1960's, there were in fact two competing interpratations of the Heisenberg uncertainty principle. The 'realist' school believed, as you do, that a particle does have a definite position and momentum, but that they are unknowable (or at least not predicted by quantum theory). The competing 'Copenhagen' interpretation, which has become the accepted one, holds that a particle does not have a definite position or momentum until a measurement is taken.
Believe it or not, this is a testable proposition. In 1965, John Bell proposed an experiment to rule the "hidden variable theory" in or out. Wikipedia explains
here and
here.
Repeated experiments have shown that the universe violates Bell's inequality and therefore does exhibit randomness.
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I'm not an expert but as I understand it Bell inequality is an attempt to show that there are random phenomena or the speed of light can be exceded.
Even if Bells test is passed (failed?) to any precision it could be that the speed limit is high enough to maintain deterministism in a way that conforms with relativity.
Imagine a computer simulation that runs much faster than the speed of light but simulates a universe with the speed of light constant as per relativity. QM could be a window into the nature of the simulation rather than evidence of randomness. [models could then be created that don't require a computer simulation].
This could all be rubbish.
chez