As I said, I understand the intention.
(1) Alice plays an Attack, which triggers all when-play-Attack abilities. Alice's Urchin, Bob's Moat, Bob's Secret Chamber, and Cathy's Moat all trigger at the same time. It makes intuitive sense (given the turn-order rule) that the triggered abilities are resolved in turn order.
(2) When Alice buys Embargo'd Noble Brigand, both abilities trigger at the same time. Here it's not so clear (which is why this thread was created in the first place). But the idea is, well Alice triggered both, so it's "her" abilities to order.
The problem is that Alice actually triggered all abilities in (1) too.
Maybe we have to look at the concept of ownership after all. The reason it's intuitive that we go in turn order in (1), is that we think of "Alice's Urchin", "Bob's Moat" etc. So in (2), the Noble Brigand is still in supply, it's just been bought, but going by the rule of Estate ownership for Inheritance, a card is actually "yours" from the moment you buy it, before you gain it. So then it makes sense that it's "Alice's Noble Brigand". But what about Embargo? Embargo isn't even a card really, it's a global rule that says that when "a player" buys a card, he gains a Curse. We could say, since it's a global rule, that it says to each player: "When you buy a card, gain a Curse." But Embargo is still not a card owned by Alice. Also, what if Bob had played a Swamp Hag and Alice buys a Noble Brigand? I assume she would still get to choose whether to gain the Curse first, but in this case the Swamp Hag is actually Bob's card. So card ownership clearly doesn't matter.
Maybe we can look even closer at the triggers. The difference between Noble Brigand's trigger ("when you buy this") and Moat's trigger ("when another player plays an Attack") is who is being referred to. Noble Brigand only talks about "you", and so Alice is both the player who triggers and the player who will be affected. Moat talks about "another player", meaning "a player other than you". This implies that the "other player" (Alice) triggers, but "you" (Bob) will be affected. But what about Swamp Hag and Haunted Woods? If Bob has played one previously, and Alice buys a card, Swamp Hag says to Bob: "when any other player buys a card". This would then imply that Bob is the one being affected, which is wrong (I assume). So this doesn't work either.
The only thing I can find that seems to work, is to define Reaction cards specifically: When Reaction cards trigger, they are resolved in turn order. But how do we define who resolves each card? It has to be either according to who each card belongs to, or according to who each each card addresses (in the trigger). (Since neither Embargo, Swamp Hag or Haunted Woods are Reaction cards, they are always resolved by the player who triggered them.) I think this would cover all interactions, but I'm not sure.
TL;DR: The reason Reactions go in turn order, is just because they are Reactions. That type sets them apart. (Otherwise Swamp Hag would behave wrongly.)
Donald, I don't know if this was your intention, that this is peculiar to Reactions? You did write: "When something happens that other players can react to, e.g. playing an Attack, they have to speak up about their interest in doing their thing, and they technically go in turn order."