Actually, the kind of grass makes a big difference. I've read that some native grasses have root systems that go way deeper than regular lawn grass. Those fancy ground covers might be shallow, but they could spread faster and cover more area quickly. Maybe it's about finding the right balance for each slope. Idk, just throwing that out there.
Have you seen what happens after a big storm? Grass roots spread out and really lock the dirt down. Those other plants often have shallow roots that can't handle the wash. I've watched slopes with fancy covers just wash away in no time. Grass might not look as nice, but it does the job better. It's all about what actually works, not just what looks good.