Why basicity decreases for leaving groups, but nucleophicity increases for an overall compound, going down the periodic table?

Question

Basically, I read that nucleophilicity increases going down the periodic table within a particular group, due to polarizability, but basicity decreases down a group (or family) in the periodic table, because those atoms can stabilize a negative charge, making them good leaving groups. Don't these two points seem to contradict each other, since increasing nucleophilicity should increase basicity?