Well, I've processed a few turks and have done it both ways. Unless you have a strong arm, keen aim, and a sharp axe, you might want to rethink that method. Sometimes if you don't strike just right, it just bounces off of a big turkey like that. Then you've got a monster bird on your hands and you are swinging until you finally hack through. I'm an ex-roofer, from the days when you used only a shingling axe to both nail and cut the wood shingles and shakes, so I'm pretty good with an axe, but have missed a time or two. It wasn't pretty, for either me or the bird.
Slitting a turk is a much more sure thing. And if it is done right, lasts nowhere near "several minutes", at least with the bird conscious. The blood and oxygen are completely cut off from the birds brain, so it makes no sense to assume they suffer for any length of time. Just make sure you cut the soft part of the neck on both sides, on the soft skin just below the bone under the ear. Make it nice and deep so the blood gushes.
We will be processing 25 BBWs over the next few days and we will be slitting.
Course, you may be already done with the deed and have likely found out for yourself, by the time you read this. Either way, good luck with it.