This card really highlights why Aggression continues to be the worst aspect in the game. This card seems hit hard by the Aggression tax.
Compare for a moment to Skilled Investigator.
Skilled Investigator triggers when any side scheme is defeated regardless of who defeated it. It costs 0. It can be played on any player so you can put additional copies in multiplayer decks, and let every player draw when a side scheme is defeated. There is no additional restriction on it aside from being in hero form.
Then we have Blood Rage. You can only use it on yourself, it costs more than Skilled Investigator. It only triggers when you kill the minion, and only if you use a basic attack. And then it deals 1 damage to you for some reason.
In fairness, when Skilled Investigator was released, player side schemes didn't exist so you wouldn't be guaranteed to have as many side schemes available to thwart. Skilled Investigator has always been really good, but player side schemes has made it much more powerful and reliable.
Protection actually has a couple cards that are similar to this. Unflappable costs the same as Blood Rage and lets you draw a card if you perform a defense without taking damage. It doesn't damage you for using it. Change of Fortune costs the same amount as Blood Rage, and rewards you for defeating enemies during the villain turn instead of rewarding you for defeating them with a basic attack. It lets you draw 2 cards for this, and doesn't damage you.
Leadership doesn't have a direct equivalent of a "draw cards when you complete a certain task" upgrade. But it does have a comparable card in Clarity of Purpose. Same cost as Blood Rage. Deals 1 damage to you. The reward is a wild resource instead of a card, which is generally worth less. But there's no restriction on when to use it. And like Skilled Investigator, copies can be given to teammates.
Technically, Unflappable and Change of Fortune can be given to teammates too. I'm not sure how often you'd want to give them to non-protection players, but it's cool that option exists for people who do, and it's weird Blood Rage can't.
Blood Rage is a "good card" in that it's the best an Aggression player has access to if they happen to be building a deck that involves a lot of minion killing with basic attacks. But it's also easily the worst of the comparable cards in other aspects.