Why is it limited to an Action card revealing cards? That makes the wording a bit weird and I don't mind it triggering on Venture.
Also setting aside allrevealed cards could cause rules issues with Horse Traders, Island, etc.
How about this:
When you reveal cards from your deck, you may set this aside from your hand. If you do, name a card and set aside all revealed copies of the named card. At the start of your next turn, discard all cards set aside this way. <-- Including the reaction card
I agree that there's no need to limit to action cards. I don't think your wording works though. For example:
I play Golem, so I am now going to reveal cards while looking for two action cards. Suppose I have Remodels in my deck but there's nothing in my hand that I want to Remodel. I reveal this reaction and name Remodel.
Intended effect: Whenever I reveal Remodel, I set it aside. I end up revealing two other Action cards, which I play as normal.
Effect of your wording: I reveal two Remodels, both of which I set aside with the reaction. But Golem still revealed two Action cards, so it stops revealing more. Not only that, it still plays the Remodels (as phantom copies, I think) even though they were set aside.
I think you need to add on some clarification at the end. And also, I think you need to say "When you would reveal cards" so that you have to use the reaction before you start revealing, or else you can wait until you reveal something you didn't want to reveal before you use your reaction to name it, instead of having to decide beforehand. This may actually be a better way to implement the card, but it's not what the OP describes.
I also don't think that this wording adequately specifies when the effect ends, which is why the OP used the awkward "while this is set aside" (which also has some possible issues -- what if something else sets it aside?).
So, maybe something like this:
When you would reveal cards from your deck, you may set this aside. If you do, name a card. Until you are done revealing cards, when you reveal a copy of the named card, set it aside and continue as if it wasn't revealed. At the start of your next turn, discard all cards set aside this way.
Note that "when you would reveal cards" doesn't prevent you from continuing on to revealing cards. Unlike Trader, this card doesn't say "instead".
And here's an attempt at that other way of implementing this concept:
When you reveal a card from your deck, you may discard this from your hand. If you do, discard the revealed card and continue as if it had not been revealed.
Or:
When you reveal a card from your deck, you may reveal this from your hand. If you do, swap this card with the revealed card and continue as if you had revealed this card from your deck instead.
Not sure if I actually like this concept overall. The interactions of this reaction might be too complicated, especially for how uncommon it would be.
