Nobody seems to think about the damage to the brain caused by the dagger stabbing through his eye into his head, causing immediate death
It doesn't cause immediate death. It can, but it usually doesn't. You'd have to be lucky and sever the right parts that control the basic body functions, which are hard to reach when you stab through the eye. Stabbing the neck would have a much higher chance to lead to a quick death (quick, but still not immediate).
As I said, in real life, people don't just receive fatal wounds and then lie still. It's a fucking dance of blood and screams, and it can last a while. Stabbing the brain can actually lead to a very slow death, slower than cutting the throat (which will spill a lot of blood quickly, dropping O2 support for the brain and leading to quick unconciousness and later brain death). If you stab someone through the eye, and don't twist the blade in the wound, the blood loss will be much slower, and thus, the victim will die more slowly and will even have a chance of surviving.