In order to have a coherent discussion about quantum mechanics, it helps to have a pretty good understanding of quantum mechanics. I think you're unintentionally mischaracterizing the arguments for non-determinism in quantum mechanics.
Let's take one of very many weird results in quantum mechanics. You have polarized sunglasses, right? What happens with polarization is that a photon has one of two polarizations which are at 90 degrees to each other; if it is aligned with the sunglasses it can pass through, but if it is aligned at right angles, it can't. If you line up two pairs of sunglasses, it's only as effective as one: the photons that were aligned with the first pair are still aligned and pass through both. Now suppose you play a nasty trick on your photons: you rotate your second pair of sunglasses by 45 degrees. Guess what happens? Half the photons go through. Half of them are aligned with the new sunglasses, and half are at right angles. It's like the sunglasses made the photon choose: did you really secretly want to be 45 degrees to the left, or 45 degrees to the right, when you said you were vertical before? And you can do this over and over and over and over. If you have vertical sunglasses, then 45 degrees, then vertical again, the photon doesn't remember anything about its first "choice"; maybe it was a "vertical, then 45 degree, then 90 degree" photon, or maybe it was a "vertical, then 45, then vertical again" photon.
At some point, saying that the behavior is determined (but by nothing that we can measure, and even with a proof that there is (probably) not anything local we could measure) starts to be unwieldy. It's not a useful description of how the world operates. After all, we say that dice are "random" when we know perfectly well that classical mechanics tell us all we need to know to calculate how dice will land given initial conditions; it's just so complicated that we never bother. Likewise, if the polarization of photons really is deterministic, it's almost impossible to imagine that it matters. Hence, the sensible scientific conclusion is that some quantum mechanical processes are random (unlike dice, we cannot determine how they will go, have no leads on finding out, and have good evidence that our normal methods will fail).
Now, back to your question: knowing that you don't know something is, of course, not a reason to say that something is false, or true, or unknowable. But when you're trying to come up with a compact predictive framework to describe observations, the theory that lets you best deal with those things you can know, and lets you adequately work around that which you don't (or can't--even if you don't know whether it's "don't" or "can't"), is the superior theory. Once that theory is shown to agree with many observations over long periods of time and under a wide range of conditions, it is pragmatic to accept that theory as true in the scientific sense (i.e. in the sense that it is "true" that you cannot fly by flapping your arms).