I watched the Echo Chamber stream that TheeChaos mentioned yesterday, and at one point they skim over the Jade Rabbit in the inventory, revealing that its stats, ammo count, and exotic perk are the same as the Fate of All Fools (although the JR has a much higher reload speed).

It's interesting, and makes me wonder if the FoAF no longer exists. It was to be a HoW weapon, but disappeared without a word from Bungie. Now we have a weapon with the same stats and exotic perk (body shots stack damage on the next headshot and returns ammo to the mag), and it's the Playstation exclusive.

notes:

Enemies get a red glow when you have them in the sights. Good way to locate snipers, but I wonder if it works on cloaked enemies.

I wonder if the chained shots carry over between reloads. The magazine size is crazy small, although other gameplay videos have shown that it's a three-shot kill in PvP. This has the potential to add a Hawkmoon-like meta, where you have to be smart about reloads. Seems odd that they'd go out of their way to remove the Magic bullet from Hawkmoon, if they were going to add it to another weapon (although, admittedly we haven't seen just how much damage can be stacked onto a single shot, but taking the Extended Mag into account...).

No Time To Explain may be what happened to the original Exotic concept. Maybe FoAF was the upgrade path for TSR at some point?