think you've raised an excellent question at the frontiers of QM research (partially indicated by your lack of answers so far), but it hasnt been entirely formally defined or captured as a problem. the question is along the lines of "what exactly can QM algorithms compute efficiently anyway?" and a complete answer is not known & being actively pursued. some of this is related to the (open questions on) complexity of QM-related classes.
this would be the case that there is a somewhat formal question defined. if the QM classes can be shown to be equivalent to "significantly powerful" non-QM classes, then there's your answer. the general theme of this type of result would be a "not-so-hard-in-QM" class is equivalent to a "hard-in-non-QM" class. there are various open complexity class separations of this type (maybe someone else can suggest them in more detail).
something strange about current QM knowledge on quantum algorithms
is that theres a kind of weird grab bag of algorithms that are known to work in QM but theres seemingly not a lot of coherence/cohesion to them. they seem odd and disconnected in some ways. theres no apparent "rule of thumb" for "problems that are computable in QM are generally in this form" despite a reasonable expectation that one could be there.
e.g. contrast this to the theory of NP completeness which is much more cohesive in comparison. it seems like maybe if the QM theory is better developed it would obtain this greater sense of cohesion reminiscent of NP completeness theory.
a stronger idea might be that eventually when the QM complexity theory is fleshed out better, NP completeness will fit "neatly" into it somehow.
to me the most general QM speedup or widely applicable strategy Ive seen seems to be Grovers algorithm because so much practical software is related to db queries. and in some ways increasingly "unstructured" ones:
Grover's algorithm searches an unstructured database (or an unordered list) with N entries, for a marked entry, using only $O(\sqrt{N})$ queries instead of the $Ω(N)$ queries required classically.