There are advantages and disavatages both ways..... I would suggest anyone who confines themselves to one or the other probably just hasn't figured out what the other is good for.
Three examples of focus priority, slow frame rate but completely in focus over (approx.) 15 frames.
I've tried this with AF-s and shutter priority, more images but not as many keepers, and definitely not 15 consecutive in focus frames in a row. Lucky to get 7.
Of course when you're using tracking like this, it's essential that you get locked on your subject. I've had frames where were my camera locked on the grass and every image was out of focus. It takes a bit more attention but when it works, it works great.
It automates the process of locking focus and pressing the shutter release. It would seem folks think they can focus, release the AF button after determining the focus has been locked faster than the camera can. I'm betting on the camera. Of course their are situations where your subject isn't moving much, you don't actually need to refocus after each image, and the faster burst speed of shutter priority gives you better odds of a keeper, if you don't run out of buffer.(which happens real quick at 8 frames per second.) And I won't settle for out of focus images. They get deleted real quick.
So maybe if you aren't shooting action, AF-s and shutter priority is better. I've done it successfully both ways, and have worked out my preference on when to use each. That's why my camera is set one way and th other way is on the dial under User1, so I can switch back and forth quickly if needed.