But there's also no question that the "when it was gained" window was still ongoing when the Skirmisher was played, as you could have definitely kept using other things that the gain triggered after that, like Sheepdogs, and Sheepdogs that you drew with other Sheepdogs after the gain.
It's not exactly the same with Reactions (and some other cards) as with Livery, Skirmisher and Priest, and also cards like Goons and Haggler that need to be in play. That's probably why some people feel like it doesn't follow from the rule about Reactions.
The reason is that the trigger on Sheepdog has been true during the entire when-gain window, even though it enters your hand during the window. It's just "when you gain a card". But on Livery, it's "this turn, when you gain a card". We know that "this turn" means "starting from now and for the rest of this turn". It hadn't started yet when the window opened. The same goes for Goons that enters play during the when-buy window ("while this is in play").
Donald even initially ruled that Livery doesn't work. Of course, that would also have meant that Hireling played at start of turn shouldn't work, and the Goons/Haggler/etc. scenarios shouldn't work, and several others. Then he reversed that ruling.
Whether you think that Livery etc. follow naturally from the Reaction rule depends on how you view the rule. You probably view it as, "things keep triggering during the window". I used to too. The other way to think of it is, "things that triggered can keep being resolved during the window". It's clear that some people see it like that intuitively. After realizing this difference, I came to think that this second view is less of a leap in logic. But in any case, it required a ruling.