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Willows are subject to a number of diseases and pests.
Crown Gall, Canker or Scab, though none of these will generally cause the bark to strip or fall off.
Poplar or Willow borers can tunnel under the bark and cause the bark to split or crack. Look for other signs of these insects, like sawdust piles at the base of the tree, swelling branches or twig dieback.
Permethrin can be used to kill these insects.
Sun scald occurs in the winter when the sun reflects off the snow. The sunlight creates heat and it heat the molecules in the tree, then when the temperature drops at night the molecules freeze.
This damages the bark and it breaks and peels.
Most trees will recover from Sun scald on there own, but watch for disease or insects in these wounds.
Injury of bark can often come from animals. Loose wire around the tree can deter critters.

https://www.gardeningknowhow.com/ornamental/trees/willow/grow-a-willow-tree.htm

I agree with you. Willows really prefer full sun. Their branches are also prone to breaking in high winds, weeping willows especially.

Here's an article all about small ornamental trees that do well in shade: https://www.gardeningknowhow.com/special/shade/small-ornamental-shade-trees.htm

They all ought to be able to stand up to the wind better than a weeping willow. Hornbeams and witch hazels in particular can tolerate moist soil.

Here's another article specifically about trees that thrive in poor, wet soil:
https://www.gardeningknowhow.com/ornamental/trees/tgen/wet-soil-trees.htm

This doesn't appear to be a vine, it looks like leaves growing from the trunk of your tree.

It looks like you have had some branch die off. Check the branches do the snap off or bend?
It appears that all the new growth is coming from the main trunk.

https://www.gardeningknowhow.com/ornamental/trees/willow/how-to-prune-willow-trees.htm

Your tree is infested with the Giant Willow Aphid.
Treatment of an insecticide will work to control these Aphids.
Here are some links with more information.

http://www.fcps.edu/islandcreekes/ecology/giant_willow_aphid.htm

https://www.ipm.iastate.edu/ipm/info/insects/true-bugs/giant-bark-aphid

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https://www.gardeningknowhow.com/ornamental/trees/willow/grow-a-willow-tree.htm
https://www.gardeningknowhow.com/ornamental/trees/willow/how-to-prune-willow-trees.htm
A balanced fertilizer with an equal ratio of nitrogen, phosphorus and potassium, such as a 20-20-20 formula, is appropriate for a weeping willow. The nitrogen supports foliage growth on the tree's long, weeping branches, while the phosphorus supports the growth of roots, stems and flowers. The potassium encourages overall health.