Liquid-crystal devices are a promising cheap alternative for terahertz light modulation, albeit they suffer from problems associated with thick cells. Here we describe a few-micron-thick polarization-independent nematic liquid-crystal metamaterial device displaying terahertz reflectance modulation depths above 23 dB, millisecond response times, low operating voltages, and a spectral tuning of more than 15%. The dramatic performance improvement is based on invoking critical coupling with external fields, which rests on a suitable choice of resonator geometry. We analyze the coupling mechanism to conclude that perfect absorption can be reached with a wide range of parameters and liquid-crystal materials. The proposed device performance, microscopic details, and the nematic molecule switching dynamics are evaluated with the use of a rigorous tensorial formulation of the Landau-de Gennes theory and shown to be robust to small parameter deviations.