Well, if their throat gets cut, then they will bleed quite much, wont they? I'm pretty sure being aware of getting your throat cut and bleeding a lot is mentally stressful.
A second or two is enough to ban it. If it takes eight seconds to reach unconsciousness and gives one second of pain, then it still causes both pain and mental stress.
Some forms of stunning do, yes, and they should also be banned. The captive bolt pistol seems to induce instant unconsciousness and be painless, while electrocution and carbon dioxide seem to be slower, more unreliable and more stressful.
Banning the slower and more unreliable methods is a start. Hopefully all methods of murdering innocent animals will be banned in the future.