Somewhere (and I cannot find it again now), I read that cutting the tree in half will do the trick. Might have to repeat. I've dug out the seedlings (pull pull pull all the roots), and just cut the six-footers. You must have the 20+ foot trees.
A Guide to Buckthorn Removal by Lisa Woolford

Many native plants struggle to survive in a buckthorn-dominated landscape. Buckthorn invades natural areas and forms dense thickets that outcompete native plants for moisture, light, and nutrients. The woody shrubs produce chemicals that make soil toxic to some animals (like many amphibians), and the leaves break down so rapidly that they expose bare soil and lead to erosion.

Additionally, at the end of summer, female buckthorns produce several hundred berries, each containing four to five seeds that can survive in the soil for three to four years. These dark berries are attractive to birds and other animals, who distribute the seeds and enhance the rapid spread of buckthorn across the region. Unfortunately, these berries have poor nutritional value and have a laxative effect for most creatures that eat them.

Buckthorn doesn’t have any natural controls such as disease or insects that would manage its growth. As it spreads and overtakes available woodland space, it dominates the territory and smothers any competition—creating a toxic environment that makes it difficult for native plants, insects, and wildlife to thrive.