I'd understand not being able to take items back because that's stealing but why would they provide tools, heck there is even a wetsuit, if you can't even take the creatures you catch back? What a tease!
The ax and shovel are definitely useful to remove bushes and trees. Some of the villages have too much in them and it's easier to make your own paths. Plus the town might be set during a time like a summer night. So having a bug net can be fun to catch the fireflies.