First off, I dont think that the hero power is (objectively) OP. 1 mana for 1 face based damage is perfectly in line with the other classes hero powers.
Where I think the design goes wrong is that the 1 mana cost makes it really flexible to weave into other turns - particularly answering 1 mana/1 health drops on turn one. No other class can do this. You also hardly ever float a mana making the class very mana efficient.
Add to this that they've printed a large number of cards which synergise very well (too well?) with your hero attacking and snowball very quickly and you start to see why it might be problematic. Battleworn Vanguard for example is basically 4/4 in stats for 3 mana - that's good in anyones book. If you cant answer it immediatly the board is full by turn 4. For me this is the crux of the issue with DH and why it feels so bad to play against (that and the fact they keep giving the class more draw unstopable face damage).
Now imagine If the hero power was 2 mana/2 damage (to keep it consistent with other classes). Things would look very different - no minion clear on turn 1, a 2 mana minion drop on turn 2 that you have to hope survives to turn 3 to activate. If you then used your hero power to attack you've only got 1 mana left which means you cant develop a good board. If you wanted to garuantee the proc it's a 4 mana play. Suddenly things look balanced!
All that said - the biggest problem at the moment is Dreadprison Glaive and the interaction with Multi-Strike. Clear a minion and get a windfury face attack - excellent!