You know that tree in our yard?

You know the tree we talk about in our back yard? The one that takes tons of maintenance? It drops branches constantly, drops copious amounts of leaves in the fall, and is a nesting ground for many birds. It also provides us with cool, much needed shade during the hot summer days.
Well the tree got a bore and several of the huge trunks started to decay and become dangerous. So we got a couple guys out to trim back the bad branches. It was a big step for me because I'm kind of addicted to the shade of the tree, even though there are many cons to having such a big, fast growing tree in a small back yard. I almost canceled at the last minute. However, I stood firm and let them start to cut away. The arborist told me a long list of reasons why it is really dumb to have a willow tree in a back yard. Ok, ok. Cut it already.
Afton and Ben watched in a trance as it took an hour to cut 2/3 of the tree down, and then three hours to clean up all the branches.
Here's the progression, for your viewing pleasure:
Phase 1: Full tree
Phase 2: One major trunk down
Phase 3: More trunks off
Phase 4: All cutting done, and just a sampling of the wreckage on the ground
Phase 5: Done!
I can't believe how much more sunlight gets into my kitchen in the morning! I actually really love it. I don't have to turn the lights on in the middle of the day anymore. We'll see how our lawn does suddenly with tons of sunlight on it. And it sure seems like my neighbors' houses are closer!
I was told the tree will grow some shoots out to even itself up a little on the left side. Hope so! For now it looks like a watermelon cut in half!
We're going to plant a new tree and let it grow for a couple years and then cut the big willow down before it uproots our foundation. Any suggestions on good, shady tree that doesn't drop sticky stuff and has nice fall foliage?