Now, the only hero this would really impact is Tinker, which is the reason for this discussion. In my last game, the enemy team was complaining because Tinker was able to jump around even though he was taking damage. This is because although damage is being dealt, he can rearm and reset the cooldown of Blink Dagger to use it.
The whole purpose of the change made to Blink Dagger goes back to the days of the name being changed from "Kelen's Dagger of Escape" to "Kelen's Dagger" because it used to have absolutely no cooldown when being attacked, you could use it like a normal blink. But this was done for reasons of balancing the item. I understand that making this change for a single hero seems a bit extreme, but that's why it's a discussion.
Personally, if a Tinker is timing their Rearm between hits of damage in order to use it, I don't see anything wrong with that.