Personally, I'd get rid of it. It is non-native, it has very weak branching structure, so tends to drop branches all the time, as I am sure you have noticed by now. As you said, it smells bad and shades everything else out. It is becoming an invasive exotic pest. It bears no edible fruit. It suckers like crazy. The trunk splits easily. They are shallow rooted and so are prone to uprooting in bad rain/wind storms. Their life span is usually only 25 - 30 years. Being so large, yours may be nearing the end of its life span any way.
They are popular street trees here in Cincinnati and I hate them. There are so many native trees you could plant that would produce actual fruit, be friendly to birds and bees and beneficial insects.