Lots of knowledgeable info here, my friends. I used to be an avid rock climber, also a (licensed) climbing guide in the Adirondacks.
I don't use a tree stand myself. if I did: yes, I'd use a rock climbing harness. I'd anchor to my full strength haul loop in back, with no slack while sitting. If rigged securely, you simply can't "fall" below the stand. Backing up the anchor with a second, both equal tension, is safer still, and standard practice for climbers.
Climbing rope is "dynamic" - as McDave has pointed out, it has some stretch to it, intentionally. The longer the length of rope to your anchor, the bore stretch you'll get. Typical elongation, is about 6-8%, I believe (it's been a while). "Static" rope has little to no give. Webbing or cordelette have no stretch. I might use either climbing rope or webbing/cordelette to anchor.
One note about anchoring in front, when you're facing away from the tree: Look at that video - the guy gets spun around to face the tree. He catches himself with foot and hand. That's without a bow in hand, in shorts, sneakers, t-shirt, in an anticipated fall. What if it were cold, lots of clothes, boots on, holding a bow with a broad head nocked on the string?
One of the scariest moments during my climbing days (my early days, I'd like to note!):
I was sitting on a ledge, anchor line running from my front loop to a solidly placed anchor behind me, belaying my partner below up a pitch.
He fell (this places the weight of the fall on your tie in point). The fall yanked me, spun me around, and smacked my face on solid rock. I held the belay, and managed to get things back together, with the help of a third partner on the ledge with me.
If I had hit harder, and/or lose the grip of my belay hand, my partner could have died.
Something to consider about your tie-in and anchor system.
Don't want to be too heavy here my friends, and certainly not preaching, just adding some real world experience to the thread.
For me, it's no slack in the system and force of fall in line with the anchor.
Be careful out there!