We employed mercury thiogallate (HgGa2S4 or HGS) to achieve the highest average output power from a 1 µm synchronously pumped optical parametric oscillator (SPOPO) generating femtosecond idler pulses continuously tunable in the mid-IR. HGS is one of the most promising wide band-gap non-oxide nonlinear crystals without two-photon absorption (TPA) losses at such pump wavelengths. The SPOPO was pumped by an 80 MHz mode-locked Yb-fiber laser with 6.5 W of average power at 1.035 µm and a pulse duration of 220 fs. Continuous tuning of the non-resonated idler was achieved from 4.4 to beyond 12 µm with an average output power of over 350 mW at 5.2 µm (>2 W at 1.25 µm for the signal) and tens of mW at the edges of the idler tuning range (hundreds of mW for the signal). The quantum efficiency of the device exceeded 20 % up to 8 µm (> 30% in the maximum). The output of the OPO was diffraction limited with a beam quality factor (M²) below 1.1. No crystal damage was observed at average power densities as high as 60 kW / cm² although using lower power densities was necessary to achieve the best possible beam quality. The only other crystals that were successfully employed in lower power SPOPOs in the past, CdSiP2 (CSP) and OPGaP, lack either the tuning range due to limited mid-IR transparency or the capability for continuous tuning.