A search for dark mesons originating from strongly-coupled, SU(2) dark flavor symmetry conserving models and decaying gaugephobically to pure Standard Model final states containing top and bottom quarks is presented. The search targets fully hadronic final states and final states with exactly one electron or muon and multiple jets. The analyzed data sample corresponds to an integrated luminosity of 140 fb(-1) of proton-proton collisions collected at root s = 13 TeV with the ATLAS detector at the Large Hadron Collider. No significant excess over the Standard Model background expectation is observed and the results are used to set the first direct constraints on this type of model. The two-dimensional signal space of dark pion masses m(pi D) and dark rho-meson masses m(rho D) is scanned. For m(pi D)/m(rho D) = 0.45, dark pions with masses m(pi D)< 940 GeV are excluded at the 95% CL, while for m(pi D)/m(rho D) = 0.25 masses m(pi D)< 740 GeV are excluded.