There are incidents where cavers are stuck on rope. The fatality that everyone remembers is that of James Mitchell in 1965(?). He was climbing out of a cave in a freezing waterfall. Even without a waterfall, cutting off blood circulation by a seat sling can be serious. When someone is stuck, the first attempt is to get him up. If that fails, he can be lowered and a better rescue effort can be organized. Another rope might be needed. The "tensionsionless" hitch allows tension to be taken off the rope gradually.
Ah, I see now. So, the "tensionless" part of this hitch has value. The tensionless working end is where you attach the additional rope. Then, you can lower the climber (your load) by either working the rope(s) around the anchor, or by twisting the anchor like a capstan if you can. I imagine using the anchor like a capstan would be rather convenient, but that option probably wouldn't be available in a climbing situation out in nature.
Another use would be pretensioning a tightrope. An ascender or Prusik would be used to pull the rope. The hitch would tie off the rope without puting any slack in it.
Likewise, the Tensionless Hitch is an option for finishing a Trucker Hitch, but I'd secure the working end somehow, probably Two Half Hitches. So, basically, we're talking Round Turns and Two Half Hitches.