Yeah this is way better than Prince (no, not strictly better. No one's saying it's strictly better). Prince requires you to 1) Buy Prince. 2) Buy the card you want to Prince. 3) Manage to get Prince and the card in your hand at the same time.
For all of that, you get the benefit of playing a $4 at the start of every turn (the same $4 each time).
Admiral requires you to 1) Buy Admiral. 2) Play it as soon as you draw it.
And for those much easier steps, you get the benefit of playing a $4 at the start of every turn, that can be whichever $4 is best for you at the moment, AND you get to gain a copy of that $4. The suggestion that gaining the card is more of a detriment than a benefit is like saying that Throne Room is better than Disciple, because Disciple could cause you to clog your deck with the card you are throning.
Yes, the fact that a pile can run out can hurt Admiral. It's not as good if there's only a single card that it works with. But even in that case, by the time you run out of the pile, you should have already gotten a LOT of benefit from it.