It does not have to mean something inherently non-deterministic or something that can't be modelled, although it certainly is the case that if something is inherently non-deterministic then it would necessarily have to be modelled randomly. Modelling things as a random process is very useful even in cases where the underlying phenomenon has a fully understood and deterministic model; a simple example of this would be chess. It's an entirely deterministic game with perfect information that is fully understood, but nevertheless all the best chess engines model positions probabilistically and use randomness as part of their search.

The use of "random" as explanation or characterization in science has certainly spanned everything from "we don't know", to "there is inherent indivisible physical randomness".

And I would agree, in the latter case it is a crutch. A postulate that something gets decided by no mechanisms whatsoever (randomness obeying a distribution still leaves the unexplained "choice").

It is remarkable that people still suggest the latter, when the theory, both in theory and experiment, doesn't require a physical choice at all (even if we experience a choice, that experience is explained without the universe making a choice).