Not only would that be harmful to the tree, but they aren't taking into account that the tree will continue to grow in circumference.

Not sure I'm visualizing exactly what you're proposing, but growing up in Louisiana, we often utilized trees (Southern Red Oak, White Oak, Southern Pines, Sweet Gum, etc) as fence posts. Usually there were no repercusions other than the barb wire grew into the trees.However, occassionally the tree dies. Not a big problem in the middle of a bunch of woods, but a bit of a hassle if the tree is near a house or electric lines, or you depended on it for shade, etc.
Not a recommended practice, but it can be done.

Another possible option, although not ideal, and requiring some creativity on the fencer's part, is to drive 2 t-posts next to the trees, attach a "false" post to them with wire, and attach the boards to the posts. It wouldn't be as sturdy as attaching to the trees, but a gap could be left, or allowed for (with future cuts with a saw), so the boards don't end up cutting into the trees or the trees growing over the boards.

If you didn't want to use t-posts, then the plastic footers they now have for foundations, with a base for a post to be slotted into, could be dug into the ground a foot or so from the tree(s), and the gap could be allowed for that way. This would need more digging than the t-posts, but....., Or you could just put the fence posts in the ground as normal, but the same foot away from the trees, and carry on as above. Both the latter options depend on how and where the tree roots grow, as cutting a major root is NOT recommended.