Doesn't the Turing halting problem also imply that (some) things are not predictable (the halting of certain algorithms)? But I don't think that interferes with causality.

Causality in physics survived the discovery of quantum uncertainty, though it was changed by it. What parts of current physics would have to be abandoned if our current inability to predict outcomes with complete precision turned out to be fundamental?

That only matters if you assume infinite measurement accuracy, and the scientific method has never assumed that.

Even if we assume the world is 100% fully deterministic, as long as our measurements are not 100% accurate we will have some amount of measurement noise which is completely indistinguishable, even in principle, from true fundamental randomness.

In fact, fundamental randomness is much easier to work around than measurement noise, since there is no risk of it being correlated with your experimental design. In contrast, measurement errors are often correlated to the measurement method, which makes them much harder to eliminate statistically.