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We achieve high detectivity terahertz sensing using a silicon nitride nanomechanical resonator functionalized with a metasurface absorber. High performances are achieved by striking a fine balance between the frequency stability of the resonator, and its responsivity to absorbed radiation. Using this approach, we demonstrate a detectivity $D^*=3.4\times10^9~\mathrm{cm\cdot\sqrt{Hz}/W}$ and a noise equivalent power $\mathrm{NEP}=36~\mathrm{pW/\sqrt{Hz}}$ that outperform the best room-temperature on-chip THz detectors (i.e., pyroelectrics). Our optical absorber consists of a 1-mm diameter metasurface, which currently enables a 0.5-3 THz detection range but can easily be scaled to other frequencies in the THz and infrared ranges. In addition to demonstrating high-performance terahertz sensing, our work unveils an important fundamental trade-off between high frequency stability and high responsivity in thermal-based nanomechanical radiation sensors.