If you need to clear your stagger every 5 seconds, you're doing it wrong. Are you being stubborn about using PB when hitting Moderate stagger? Sometimes that's not ideal, far from it. It's a balancing act, and you need to adjust your PB usage according to the fight you're on.
For example, on Feng, you can clear for every Moderate Stagger no problem since it's a low tank damage fight. On the other hand, on a fight such as Wind Lord Mel'jarak, if you try to clear ever Moderate Stagger then you're going to run out of Chi incredibly quickly. You're much better off waiting for Heavy Stagger and clearing that, and leaving Moderate Stagger up. Overall, that will mean you take less damage.
It's a balancing act and I absolutely do not want to see it changed. I love the stagger mechanic as it is.
I think more stuff could give you Shuffle, but I don't think increasing the duration of the Shuffle provided is the right way to go (as per a previous post of yours). As it is currently I am able to keep Shuffle up nearly 100% of the time when tanking. The only exception is during heavy AoE and I want to use Breath of Fire, which doesn't provide Shuffle. So really, I end up not bothering with Breath of Fire, instead using SCK to keep aggro and using BoK to keep Shuffle up. This is an absolute shame because BoF is such an iconic Brewmaster ability. We absolutely should be using it, but as it is currently we can't.
I've commented on BoF before and my ideas prior were to make it do more damage so you'd essentially choose, even on single target: do you want to mitigate more damage (BoK) or deal more damage (BoF). Problem is, as a Tank that's never a choice: if you don't choose to mitigate damage, you're doing it wrong. The choice instead should be in how you mitigate that damage, e.g. the choice between using PB more often or using BoK more often. That's good, interesting choice.
As such, to bring it back to your own point, I would add Shuffle to BoF, same duration as BoK (6 seconds). I'm fine with the 6 second duration, I find that at just the right level of not too demanding to keep up, but you still have to be aware of it. Even just adding 1-2 seconds of duration to each application of Shuffle would make it far too easy to keep up.
I would also like to see further changes to maybe allow BoF to be used on single targets too, but I'm unsure of how to go about doing that. I wouldn't want to replace BoK entirely, but I do want to use BoF. As I said, it's an iconic Brewmaster ability and very cool, it's a shame to see it mostly unused.
But then I can't hax the pull on Lei Shi and ignore the first Spray cast entirely while I get into position!
Nah, I'm actually ok with the aggroing of the statue. It just takes a GCD to put it up early in the fight, not really a big deal. I can also imagine it being used to easily position bosses in certain places at the pull... Not important for any boss currently (that I'm aware of), but it may be important later on. Who knows.
I mean, it's thematically appropriate, no? I mean, we're tanks. We're meant to be all about the aggro.