Results are reported from a search for new physics beyond the standard model in proton-proton collisions in final states with a single lepton; multiple jets, including at least one jet tagged as originating from the hadronization of a bottom quark; and large missing transverse momentum. The search uses a sample of proton-proton collision data at $\sqrt{s}=$ 13 TeV, corresponding to 137 fb$^{-1}$, recorded by the CMS experiment at the LHC. The signal region is divided into categories characterized by the total number of jets, the number of bottom quark jets, the missing transverse momentum, and the sum of masses of large-radius jets. The observed event yields in the signal regions are consistent with estimates of standard model backgrounds based on event yields in the control regions. The results are interpreted in the context of simplified models of supersymmetry involving gluino pair production in which each gluino decays into a top quark-antiquark pair and a stable, unobserved neutralino, which generates missing transverse momentum in the event. Scenarios with gluino masses up to about 2150 GeV are excluded at 95% confidence level (or more) for neutralino masses up to 700 GeV. The highest excluded neutralino mass is about 1250 GeV, which holds for gluino masses around 1850 GeV.