The only tasks that come to mind that involve a moving robot are tasks that require moving around to perform the task. A Roomba is the obvious and most well-known example, but there are also now robotic lawnmowers that make sense for the same reason (and I even use the city of Halifax using them earlier this year to maintain public grounds!).
Another use case I can think of that physically requires moving around would be watering plants. That would require a more advanced and larger robot, so it's not there yet, but imagine a Roomba-type base station but with a water hook-up, and a robot with a water tank and an extendable hose-arm. For how many house plants I have, how much work it is to water them regularly, and how much better the robot could do the job if it watered plants daily (but only exactly the amount it needed by using a moisture probe), this is definitely something I would consider.
The problem of course is that all of these actually useful robots are purpose-specific; none are general. A Roomba is not a lawnmower is not a plant waterer. The Amazon device tried to be a general purpose robot, but because it doesn't have vacuums or grass cutters or a water hose or anything used to actually accomplish a specific task, it's good for nothing. I can walk aimlessly around my apartment too, but I don't need a robot for that!
Minor nitpick, I've just come back from a multilevel house with stairs and ways to roll to|from any room; wide doorways, ramps, spiral outside groundwork, screw-drive elevator.
It's well thought out and practical, the own builder did the shed first and lived on site for two years with earth moving gear (owner | contractor) digging out big rocks, placing them, recessing walls into hillside, building out steel frame, etc.