Stockpile creates an unfortunate incentive to keep your mana open. I feel if you got a chance to play it "until the end of your next turn", the mechanic would play better.

I did take for granted that the expectation that the player will at least consider keeping their mana open to cast the stockpiled card.

There are two issues with your suggestion.
1.That is a very red ability, and stockpile is supposed to be exclusively a WU mechanic.
2.The general hope was that the stockpiled cards are not cards one would want to cast immediately. "Until end of next turn" would go in the opposite direction of the feel the mechanic is going for.