Look at it this way:
We can't be sure why Arias brought the gun. I would tend to think it was out of the adage "better to have it and not need it than need it and not have it", due to whatever "plan" she had for the murder itself not really being solid in her mind, or to deal with inadvertency, witnesses or the possibility of taking out someone else if Travis was on the couch with another women.
But in the actual murder itself none of the wounds were immediately fatal. The stab to the inferior vena cava and pericardium would have been eventually fatal, but not immediately. The stab wounds to the back and slashes to the head were mostly superficial, being stopped by bone. The most devastating and fatal wound was the throat cut.
Now, there's really no subtle way to put this: Cutting people's throats and videotaping it has become a preferred terror tactic. They did it in Iraq, during the Chechen War and Mexican drug cartels do it. Many of these videos have ended up on the internet. I've seen a lot of them.
No one ever dies easy this way. In all the videos, even when a person's throat is cut down to the spine relatively quickly resulting in unconsciousness within 30 seconds or so, unconscious doesn't mean dead. It doesn't mean peaceful either. Even with the blood flow to the brain interrupted, the body still struggles to breathe. In the end it may come as "agonal breathing" where maybe a handful of reflexive breaths are taken in a minute. Before that, the body will work at breathing, sometimes almost hyperventilating even though the person is unconscious. On top of that, the breathing is done with copious amounts of blood flowing into the sectioned trachea, the victim is essentially drowning in their own blood. The body will struggle, writhe, fight in a purposeless manner to continue autonomic functions. This will continue until the person doing the cutting works their way through the vertebrae and cuts the spinal cord, which is no easy feat with a knife, or until the dying person runs out of steam in a minute or two...or three.
It's not pretty. It is, in fact, horrific. That is why cartels, for example, videotape it to show what happens to rival cartel members and others that would cross them.
I don't think, at the moment of the head shot, that Travis Alexander was dead. He was mortally wounded, but the "post mortem" appearance of the head wound was due, in part, to there being essentially no blood flow to the head.
If this was the case with Travis Alexander, that once his throat was cut his body still struggled to stay alive, and I have no reason to think that the throat cutting of Travis Alexander was any different than the sample I've seen, then Jodi Arias resorted to the gun as a coup de grace, to finish it because, despite what she saw in movies, killing someone is a lot harder and takes longer than she counted on.
