I would second Anke's goats are not low maintenance and don't love grazing.
Geese have a good go at keeping a lawn nice, (if it starts to grow long in May/June they stop liking it so much so might need help from a mower sometimes). They also poo, it's not too offensive but not great if you want to walk about in sandals. They'd need to be shut away at night, but I wouldn't be too worried about fox in the day time (they are quite formidable especially Spring/summer - which is why I got rid of mine as they don't mix with small children!).
I heard on radio 4 gardeners question time a few years ago that a herd of free range Guinea pigs make excellent lawn mowers. Have to shut them away at night and give them several shelters around the lawn so they can hide from predators (cats, foxes, raptors, etc.) but think it was one of Bunny Guinness' lecturers or someone who reckoned they were just the job for keeping turf perfectly short! They don't worry other garden plants or dig, just graze grass and look cute. They're poo isn't offensive (unlike something like pigs). In winter they'd need proper housing/runs/garage and hay/veg. She didn't state stocking density, sorry.
I have a bit of a fantasy about keeping some once we get the garden fenced in.