A search for long-lived, massive particles predicted by many theories beyond the Standard Model is presented. The search targets final states with large missing transverse momentum and at least one high-mass displaced vertex with five or more tracks, and uses $32.8\text{ }\text{ }{\mathrm{fb}}^{\ensuremath{-}1}$ of $\sqrt{s}=13\text{ }\text{ }\mathrm{TeV}$ $pp$ collision data collected by the ATLAS detector at the LHC. The observed yield is consistent with the expected background. The results are used to extract 95% C.L. exclusion limits on the production of long-lived gluinos with masses up to 2.37 TeV and lifetimes of $\mathcal{O}({10}^{\ensuremath{-}2})\ensuremath{-}\mathcal{O}(10)\text{ }\text{ }\mathrm{ns}$ in a simplified model inspired by split supersymmetry.