Villagers can't pick things up, fish, catch bugs, or dig so you're good in regards to them.
They can hold the tools but it's only to imply they can do those things. Even though they can't.
I find it funny how they can turn in bugs for the Bug-Off contest and yet, you never see them catch anything. At most you'll see them attempt, but they never swing their net. Maybe they can catch bugs, but they're just too shy to do it in front of you?
3DS FC: 4141-2123-1945 Currently Playing: Animal Crossing New Leaf and Zelda Oracle series.