My brief online search tells me weeping willows can grow to 50ft talk and 50ft wide. Is there a smaller variety that would only be 20-30 feet wide? Or a similarly inviting shade tree that is smaller than a weeping willow?
This link is to a list of small trees for the Seattle area. What you won't see on there are willows. Willows have an aggressive root system and tend to be always dropping leaves and branches.
If you want a tree to sit under you want one that does not shed branches or fruit on you. Wasps and other wildlife are attracted to ripe fruit so I would skip the crab apples.
They list a magnolia but I find that they look great for three weeks and then are dull for the rest.
They also recommend a serviceberry which is a great tree but slow to grow and not very long lived.
The best choices from the list are the smaller maples like Acer ginnala, Eastern Redbud (Cercis canadensis) and the Pagoda Dogwood (Cornus alternifolia).
The last part of the choice is what is available. Start visiting tree nurseries and look at what they have and what shape it is in. No point in buying a tree that has been dried out over the summer. Pass over the ones with poor structure as this defect will not correct itself and can be a cause of failure in later years.
If you want a weeping tree, you can look at weeping cherries as others have mentioned in the comments. As kevinsky mentioned, any of the fruit trees will attract bees in the spring and wasps when they're in fruit, but personally I've never minded this fact. If you keep the fruit cleaned up, I don't think it is generally unmanageable, and you can always just choose to sit somewhere else when it is in fruit.
There are also a few varieties of weeping beech that are absolutely gorgeous and would be small enough for the space you described. Be careful, though, because there are some varieties of weeping beech that would get too large!
Another option would be a weeping hazelnut, but they may be smaller than what you are looking for - about 10 feet tall at maturity.