We experimentally demonstrate an efficient and broadband extreme-ultraviolet light (XUV) out-coupling mechanism of intra-cavity generated high harmonics. The mechanism is based on a coated grazing-incidence plate (GIP), which utilizes the enhanced reflectivity of s-polarized light in comparison to p-polarized light for large angles of incidence (AoI). We design and produce a 60°-AoI coated GIP, tailored specifically for the high demands inside a sub-50-fs Kerr-lens mode-locked Yb:YAG thin-disk laser oscillator in which high harmonic generation (HHG) is driven at ∼450 MW peak power and 17 MHz repetition rate. The coated GIP features an XUV out-coupling efficiency of >25% for photon energies ranging from 10 eV to 60 eV while being anti-reflective for the driving laser field. The XUV spectra reach up to 52 eV in argon and 30 eV in xenon. In a single harmonic, we out-couple 1.3 µW of XUV average power at 37 eV in argon and 5.4 µW at 25 eV in xenon. The combination of an improved HHG driving laser performance and the out-coupling via the coated GIP enabled us to increase the out-coupled XUV average power in a single harmonic by a factor of 20 compared to previous HHG inside ultrafast laser oscillators. Our source approaches the state-of-the-art out-coupled XUV power levels per harmonic of femtosecond enhancement cavities operating at comparable photon energies.