Here's my explanation. I don't know if it'll help anyone understand things better, but maybe.
It may help to imagine that Duplicate and Sheepdog have an implicit "while they're in a certain zone" in their text, because how else are you using that ability. So these two technically have the following text:
Sheepdog:
+2 Cards
While this is in your hand, when you gain a card, you may play this from your hand.
Duplicate:
Put this on your Tavern mat.
While this is on your Tavern mat, when you gain a card costing 6 or more, you may call this, to gain a copy of that card.
These cards basically work the same way as Groundskeeper and friends. The only requirements are A) they're in a certain zone, and B) you gained a card. (Ok Groundskeeper and Duplicate has other requirements for the gained card but whatever.) So if you gain a Duplicate and Innovate it, both conditions are met, so you can call it.
Livery has a different requirement though. Instead of being in a certain zone, the card needs to be gained after the ability has resolved. Is that clear from the wording? I dunno, that's subjective. No casual player is coming across these Innovation edge cases anytime soon. Anyways it's what Donald X. has ruled. So if you Innovate a Livery, it fails the condition and doesn't gain anything.
Oh yeah there's another case: playing Hireling at the start of your turn. It has "for the rest of the game." So why does that work, and Livery doesn't?
"For the rest of the game, at the start of each of your turns" matches Livery, in terms of this "What's the scope of this" issue. A difference though is that "the start of a turn" is less like a thing that happened that could have been missed; it's this time period when stuff happens, more like a start of turn phase, even though it's not phrased like that and I mean isn't that.
Alright that's basically the summary of this discord conversation in April (including the new Merchant ruling). I think there was another edge case involving Way of the Mouse but man screw that card, you will not drag me into that conversation.