I think if Order of Masons shenanigans means that you shuffled nothing, then that should also apply to Astrologers, Star Chart, and Stash too. It seems clear to me that you're not shuffling the Stash, you're just Secret Passaging it into the cards you're shuffling.
Unlike Order of Masons, Stash creates a new deck.
When you shuffle, you may put this anywhere in your deck. (Stash 1st Edition)
1E Stash is repositioned after shuffling. Is 2E different? That is the question.
It's true that Emissary should work the same for Star Chart, Order of Masons and Order of Astrologers.
I always thought of Stash as triggering after you shuffle. You take out the Stashes and reposition them. Stash said "when you shuffle", so that made sense, and worked technically.
(Stash v.2 says "when shuffling," but the card is not supposed to have changed functionally. Star Chart says "when you shuffle", the Orders say "when shuffling". I don't think these wording differences mean anything.)
Then Star Chart came out. It had to trigger before shuffling, unlike Stash. If you use it then (before shuffling), picking a card, it sets up the effect that you place the card on top after shuffling.
The Orders are of course similar to Star Chart.
Stash can also be interpreted in the same way. For Stash it makes no difference, both ways work (unlike for the other 3 cards). But now with Emissary, of course it does make a difference when Stash triggers.