Yesterday I stopped by a small store whose motto is "We are two small to carry junk." Turns out the guy working there is a hanger. I'd just gone to get a couple caribiners.
He is also a mountain climber. He said "Never attach like to like materials." There goes my idea to not use metal.
He suggested hanging with straps used for attaching kayaks to the roof of a car. The don't stretch at all, are very strong and easy to use. A 7 foot length is $5.95 and weighs 120 grams.
Loop it twice around the tree and attach the caribiner from the hammock to just one loop so the weight tightens the strap. It completely eliminates the minor slips of another strap on the tree, getting a bigger banana as nylon stretches, and little creaking from knots slipping. I use a structural ridgeline, but attach it after I hang the hammock and tighten it with a trucker's knot.
I'm still new at this, but this is the solution I'm going to go with for awhile. I do have a new tarp on the way.
My old way is on the other side.