This is related to Matt Satriano's earlier question about an analog of Artin's theorem for stacks with an fpqc cover by a scheme.
Suppose one developed the theory of stacks in the fpqc topology and by fpqc-algebraic meant only (a condition on the diagonal plus) "admitting an fpqc cover by a scheme" (rather than the usual definition of algebraic, which I'd like to call fppf-algebraic). One seems to give up the ability to make sense of notions that can be checked locally on smooth covers but not on fpqc covers. But some notions still seem to survive for fpqc-algebraic stacks (e.g. "locally noetherian," cf. EGA IV 2.2.14).
Question: do people principally work with fppf-algebraic stacks rather than fpqc-algebraic stacks simply because one gets a nicer package that still applies to most examples one comes across in practice?