Doesn't drilling a huge hole in a tree kill it

No, anyone with a simple understanding of arboreal anatomy knows that drilling a hole into or even through a healthy tree will not, in and of itself, kill the tree. For most trees, the majority of the core (the dry heartwood and the wet sapwood) is comprised of the rings of former growth that act as structural support and sap storage. The living part of a tree that carries nutrients and energy up and down between the roots and the needles/leaves is between the dead core and the equally dead bark. As long as this very thin living layer of the tree is not damaged all the way around the circumference of the tree, nutrients and energy will still pass.

And the size of the hole can be relatively substantial. There are places in California where you can drive a car through a living redwood tree.

The best way to passively kill a tree is to "ring" it as the Native Americans did - strip off the bark and the vascular cambium layer of living cells all the way around the tree in a wide strip. Many a tethered dog has killed a tree by way of a wrapped chain using this method.

Drilling a huge hole into a tree and leaving the hole empty does invite damage (from critters, water, fungus, etc.) that can hollow out and destroy the interior integrity of the tree over time. Pete & Co. tap the trees responsibly. They then fill the holes with non-reactive anchors because something like a cooper bolt introduced into these holes would surely poison the tree.

So no, the holes and equipment that they put into the trees on this show are really not harming the trees. The crew does not like to anchor the houses to the trees in so many places, basically encircling the tree, that they risk cutting off the cambium layer and definitely killing the tree.