I have serious doubt about how much luck affects anything. For one thing, I am pretty sure I saw a kid with luck in love. For What!? And phobias, fears and likes seems to mean nothing as well. I catch bugs that my mayor is afraid of without any problems.

I did get the rabbits foot, and I try to have a mayor lucky in "finding" just in case.. But I have had some with other luck, and as far as I can tell there has been no difference..