Metallic spintronic terahertz (THz) emitters have becomewell-established for offering ultra-broadband, gapless THz emission in a variety of excitation regimes, in combination with reliable fabrication and excellent scalability. However, so far, their potential for high-averagepower excitation to reach strong THz fields at high repetition rates has not been thoroughly investigated. In this article, we explore the power scaling behavior of tri-layer spintronic emitters using an Yb-fiber excitation source, delivering an average power of 18.5 W (7 W incident on the emitter after chopping) at 400 kHz repetition rate, temporally compressed to a pulse duration of 27 fs. We confirm that a reflection geometry with back-side cooling is ideally suited for these emitters in the high-average-power excitation regime. In order to understand limiting mechanisms, we disentangle the effects on THz power generation by average power and pulse energy by varying the repetition rate of the laser. Our results show that the conversion efficiency is predominantly determined by the incident fluence in this high-average-power, high-repetition-rate excitation regime if the emitters are efficiently cooled. Using these findings, we optimize the conversion efficiency and reach highest excitation powers in the back-cooled reflection geometry. Our findings provide guidelines for scaling the power of THz radiation emitted by spintronic emitters to the milliwatt-level by using state-of-the-art femtosecond sources with multi-hundred-Watt average power to reach ultra-broadband, strong-field THz sources with high repetition rate.