Fiber-to-chip coupling is an essential issue for taking high-performance integrated photonic devices into practical applications. On a thin-film lithium niobate platform, such a high-performance coupler featuring low loss, large bandwidth, and polarization independence is highly desired. However, the mode hybridization induced by the birefringence of lithium niobate seriously restricts a polarization-independent coupling. Here, we propose and experimentally demonstrate a high-performance and polarization-diversity cantilever edge coupler (EC) with the assistance of a two-stage polarization splitter and rotator (PSR). The fabricated cantilever EC shows a minimal coupling loss of 1.06 dB/facet, and the fully etched PSR structure shows a low insertion loss (IL) of -0.62 dB. The whole polarization-diversity cantilever EC exhibits a low IL of -2.17 dB and -1.68 dB for TE0 and TM0 mode, respectively, as well as a small cross talk of <-15 dB covering the wavelength band from 1.5 µm to 1.6 µm. A polarization-dependent loss <0.5 dB over the same wavelength band is also obtained. The proposed fiber-to-waveguide coupler, compatible with the fabrication process of popular thin-film lithium niobate photonic devices, can work as a coupling scheme for on-chip polarization-diversity applications.