I like your idea of adding a "sleeve" around the line where it wraps around the tree. It's not quite a strap, but it can take the friction, kind of like wearing two pairs of socks. I'll have to give that a try too.
The DutchWare version that I illustrated and demo'd is a bit tricker to adjust than the two mini carabiner version. With the two carabiners, you can leave the line fairly slack yet the tarp won't fall or slide off to the ground. Sliding has minimal if any friction on the tree in my experience.
Another thing to try with the DutchWare version would be to leave a little slack in the line and do a quick wrap around the "antenna" of the Stingerz/Tarp Flyz (depending on what version you use). This would lock the line in place, just like in the carabiner example, but leave enough slack in the line so you can still easily slide the ridge line (if necessary) to center the tarp. Once centered/positioned as desired, unhook the antenna, cinch the Tarp Flyz and wrap to secure.
I'm going to have to give this a try, but I think this is the answer I was looking for. Thanks for letting me ramble off an idea.
I may have to incorporate that into the video...