I'm just stating as far as I know. Maybe there're truly monks who practice "Wudang Kung Fu" and Taoism. But as far as I know, Taoism is about living peacefully and healthily and as long as possible, not about defeating your enemy.
The arts passed down from Wudangshan is not exclusive to Tai Chi. Tai Chi has origin myths relating to Wudangshan, but I doubt taijiquan was the only thing they practiced. Taijiquan itself cannot be characterized as "a slow sport to help find inner peace". Well, you can, but that is like saying "Wudang Kung Fu".
There are definitely "Taoist monks". They have their own temples and do their own things. They have similar beggar-monk tradition as Buddhist beggar-monk traditions. At several points, Taoist monks cultivated within the same spaces as Buddhist monks. This is not surprising given that all religious wisdom springs from the same source.
One aspect of Tao-"ism" is about living peacefully and healthily and as long as possible ... but that's not really it at all. That's the popular religion. As Joseph Campbell noted, the most popular religion in the world worships the idols of Longevity, Prosperity, and Posterity. Pop Taoism is no exception.
After you shed the outer layer of pop Taoism, at Taoism's core, it shares the same underlying understanding of reality as Buddhism, Shinto, and even the obscure parts of Christianity, Judaism, and Islam. Its moral expression is uniquely Chinese.
As for "not about defeating your enemy", this goes into something very interesting, the role of violence and spiritual growth. Sun Tzu's Art of War is very much Taoist-flavored. It might be kept at its arm's length, but its far more obscure inspiration, Master of Demon Valley (鬼谷子) is acclaimed as part of the Taoist tradition, despite being heavily encrypted teachings about the shadow side, governance, and right action.