The β-delayed neutron-emission probabilities of 28 exotic neutron-rich isotopes of Pm, Sm, Eu, and Gd were measured for the first time at RIKEN Nishina Center using the Advanced Implantation Detector Array (AIDA) and the BRIKEN neutron detector array. The existing β-decay half-life (T 1/2) database was significantly increased toward more neutron-rich isotopes, and uncertainties for previously measured values were decreased. The new data not only constrain the theoretical predictions of half-lives and β-delayed neutron-emission probabilities, but also allow for probing the mechanisms of formation of the high-mass wing of the rare-earth peak located at A ≈ 160 in the r-process abundance distribution through astrophysical reaction network calculations. An uncertainty quantification of the calculated abundance patterns with the new data shows a reduction of the uncertainty in the rare-earth peak region. The newly introduced variance-based sensitivity analysis method offers valuable insight into the influence of important nuclear physics inputs on the calculated abundance patterns. The analysis has identified the half-lives of 168Sm and of several gadolinium isotopes as some of the key variables among the current experimental data to understand the remaining abundance uncertainty at A = 167-172.