If knife defense is taught correctly, it is invaluable. There are two fundamental rules:
1. You're most likely gonna get cut. Minimize the damage you take, maximize the damage you give.
2. If in doubt, see rule #1.
Remember that the attacker's focus is on the knife and his mindset is that he is superior because he is armed. Once you break his focus, you gain the advantage and you don't stop until he's on the ground and incapable of fighting with you any more. He's presented deadly force against you, so he's bought and paid for in my book. Anything you do is justified, up to and including lethal force in return.
I would add "control the weapon" as a fundamental rule. I agree with you on the mindset, though. On the occasions that I've been faced with a knife, the attacker has completely ignored their other limbs. They would struggle to regain control of the knife hand, but they never even tried to strike with their other hand. I've seen this in training as well. The weapon becomes THE weapon, to the exclusion of other options, to most people.