High-precision mass measurements of eight neutron-rich nuclei in the $N=126$ region below ${}^{208}$Pb have been performed using multiple-reflection time-of-flight mass spectrometry at the FRS Ion Catcher, GSI. The measurements include the first mass measurements of ${}^{204,205}$Au and the first direct mass measurement of ${}^{207}$Tl. The new masses reveal an apparent anomaly in the evolution of binding energies along the $N=125$ and $N=126$ isotonic chains. It is demonstrated that the $N=126$ anomaly is a consequence of a reduced valence proton-neutron interaction strength $\delta V_{pn}$ in ${}^{205}$Au and ${}^{206}$Hg nuclei, with an analogous interpretation applying to the $N=125$ case. Furthermore, the first measurement of the isomer-to-ground-state ratio of the ${}^{205m}$Au $(11/2^{-})$ isomer produced by projectile fragmentation, and an upper limit of 0.5% for the isomer-to-ground-state ratio of a predicted long-lived isomer of ${}^{204}$Au are reported. These findings underscore the importance of mass measurements in the $N=126$ region for constraining theoretical models and may guide future studies of r-process nucleosynthesis.