Emergence of various circulating SARS-CoV-2 variants of concern (VOCs) promotes the identification of pan-sarbecovirus vaccines and broadly neutralizing antibodies (bNAbs). Here, to characterize monoclonal antibodies cross-reactive against both SARS-CoV-1 and SARS-CoV-2 and to search the criterion for bNAbs against all emerging SARS-CoV-2, we isolated several SARS-CoV-1-cross-reactive monoclonal antibodies (mAbs) from a wildtype SARS-CoV-2 convalescent donor. These antibodies showed broad binding capacity and cross-neutralizing potency against various SARS-CoV-2 VOCs, including B.1.1.7 (Alpha), B.1.351 (Beta), P.1 (Gamma), and B.1.617.2 (Delta), but failed to efficiently neutralize Omicron variant and its sublineages. Structural analysis revealed how Omicron sublineages, but not other VOCs, efficiently evade an antibody family cross-reactive against SARS-CoV-1 through their escape mutations. Further evaluation of a series of SARS-CoV-1/2-cross-reactive bNAbs showed a negative correlation between the neutralizing activities against SARS-CoV-1 and SARS-CoV-2 Omicron variant. Together, these results suggest the necessity of using cross-neutralization against SARS-CoV-1 and SARS-CoV-2 Omicron as criteria for rational design and development of potent pan-sarbecovirus vaccines and bNAbs.