Alas, this does not work. That is because the damage part is part of the activation. Comparing this with city of brass:
Whenever City of Brass becomes tapped, it deals 1 damage to you.
Tap: Add one mana of any color to your mana pool.
you will see what the difference is. Killing your opponent using the icy on a city will work. Combining it with the path is a little bit more difficult.