But then I stated that random, as the conceptual counterpart of the dualism random/predictable, doesn't really have relation to "objective existence" save through us as observers, and that we therefore cannot really say that our science applies to "objective existence".
That's pretty much of a strawman argument. Nobody here was defending the notion of "objective existence," which isn't surprising because it isn't a terribly useful concept.
It appears to me (and correct me if I'm wrong) that you want to limit the concept of randomness to "observed randomness," setting it apart from "objective randomness." For you, "observed randomness" is all that we can perceive, and thus "randomness" is a function of observation.
As I pointed out above, that would be a pretty radical concept -- if this were 1740. The idea that induction (or empirical observation) cannot "prove" anything was revolutionary when David Hume came up with the idea in his Enquiry Concerning Human Understanding
, but it's universally accepted today.
As for your notion that "randomness" is simply a function of observation, the same could be said for practically anything. For instance, if we define a "tree" as "a woody perennial plant having a single usually elongate main stem generally with few or no branches on its lower part," we need not add "as it would appear to an observer," since that is assumed (after all, we can't know what an unobservable tree would look like). Adding "as it appears to the observer" to the definition doesn't really add anything to our understanding of the concept -- indeed, it was always there to begin with, implicit in the definition. Your discovery, therefore, that randomness is a condition of observation is a bit like discovering that the earth revolves around the sun. Been there, done that.