Willows, and particularly Weeping Willows, are the dirtiest, nastiest, brittlest trees around. They tend to break very easily in bad weather and, if you are lucky, it will just be the small branches that fall on your house. When you do cut them down, you get nice rows of mushrooms where the roots have rotted just under the soil. You do not get usable firewood for at least a couple of years because the wood is so full of water that it will not burn for a while.
Based on these comments, what do you think should be done.