You will need to utilize herbicide treatments to effectively eliminate established willow trees.

Mossy Oak - Willow Article

They can be controlled with a foliar treatment (spraying the leaves and the whole tree is easier for many people). The linked article describes one way to combat the trees.

I generally have the best results using cut stump herbicide treatments on invasive trees at our farm. IMO, you get more herbicide to the roots and less out into the adjacent areas. Doing the treatments during the fall while the trees are sending sap to the roots also increases the efficacy.

If you are absolutely opposed to herbicides (even though they make "pond safe" products), willows can be killed by proper girdling. This will kill you big ones, but you probably won't be able to get the small stems this way. You will have to make willow tree management an annual task on your pond.

We only have a few willows at our farm and they are not invasive - so I do NOT have experience killing pond edge willows. Hopefully, someone with lots of treatment experience will drop into your thread.

Good luck on getting your pond back into your preferred condition!