No, it doesn't. I have been using my HH 1 for a long time now with no advancing tether but still with a small corner trap and it works great. The dynamics completely change with a slack tending pulley set on the carabiner ring interface. When pressure is applied, the HH will run straight up the rope without flopping over, it feels just like the Hitchclimber. The same is true when using knee and foot ascenders under the HH. Clipping into the bridge ring with a chest or shoulder strap causes the HH to run straight up the rope without flopping over so there is no additional set back.
The HH 2 does all of this but better and does it without a corner trap.