To each his own. I say trim it back well and keep it. If that tree is the northern one of two, then your house faces west. That tree certainly gives you good shade in the afternoon. In my opinion that has significant value, just not easy to put a price on.
If the root system is attacking the foundation, or sewer line, then you may have other issues to deal with that may steer you toward removal.
Whatever you ultimately decide to do, someone will think you should have gone the other way.