I have mixed feelings about this. Neighbor passed away 5 years ago from ALS, and we take care of the 75 year old widow's lawn as well as our own. Honestly, sometimes it's hard to make time for both lawns. But she's a good friend, and we promised the (now-deceased) husband we would do this for them. On the other hand, it's sometimes the only exercise I get for the week, so I'm appreciative of the "requirement." That said, I do wish we could get rid of (read: plant a garden there) one especially treacherous slope. Then things would be much easier.
We have that kind of slope at our place, in 2 spots. What we did:
- For the slope we have to walk on: plant creeping thyme. It can be stepped on (well, once it matures a bit, don't step on baby plants!) grows about 4 inches high as total groundcover, gets little pink/purple/white flowers, smells amazing when you step on it, and doesn't need mowing. Looks great, low-maintenance, not super expensive to put in. Also you can go filch some for cooking, as long as it hasn't been sprayed with something.
- For the place we don't have to walk on, that is also in near-total shade so the grass is continually scraggly and muddy: creeping jenny. The dark green kind. It is INVASIVE, so do NOT plant it outside a container garden or in a place where it can spread. WARNING. That said, mowing AROUND it seems to keep in in check, so we've got it against the house in the 5-foot patch where grass wouldn't grow, and then mow around the edges where the grass starts, and that's keeping it in check fine. Way better than muddy soil and half-dead grass...