That's of course true (In fact I tend to also believe in a non-deterministic universe "at the core").
But if determinism falls out in the end, it's still a hint that there may also be deterministic effects at the root. Current observation can't rule that out, it's just our model which assumes pure randomness. But there are lot's of possibilities how randomness can sneak in into QM which doesn't contradict obserservation.
And unless we solve the measurement problem in QM (by finding a unified theory from which both Schoedingers equations and Borns rule can be derived), it's still an open question. So considering it solved today is quite premature.
> ... chaotic systems ...
That's still deterministic. Sure, there may be some influence from quantum effects which then are amplified, but the dynamic of the chaotic system itself is still deterministic.
> (SR) ... predicting an identical speed of light for all observers
That's not really true. "identical speed of light for all observers" is an observation which was replicated quite often. SR is a way to explain this observation, but there before SR Lorenz already had a different model explaining it too. SR won, because Lorenz used an (at the time) unobservable "ether" and Einstein argued that its better to use Occams Razor and throw this "ether" away.
But Einstein didn't now about QFT, the Big-Bang and the microwave-background - which all contradict Einsteins assumptions: QFT uses an "ether-like" vacuum, the Big-Bang created a "T=0" for the universe and with the microwave-background also an absolute reference frame for an absolute time. This in all contradicts SR, so maybe SR is really wrong on a global level.
Which in turn would allow a non-local, realistic interpretation of quantum measurements because without SR simultaneity could be back on the table.