Many years ago I tried splitting focus from the shutter because I was shooting action-sports. It took about a day to decide I liked it.
One of the things we're supposed to do when we're shooting is keep both eyes open; That helps avoid fatigue from shooting for hours, but also lets us see what is going on around us. That is smart in case good action is happening to the side. It's also good because you might need to be aware of an unsafe condition unfolding while you're shooting.
Anyway, after many years of using that sample configuration on my bodies, one of the things I enjoyed again and again was the ability to track some action, see someone or something about to get in the way, release the focus button so it wouldn't mis-track, then press it again as I panned past the intervening object. It was REALLY useful.
Otherwise, being able to tap the focus button to pre-focus was great. It also worked really well when using a tripod to shoot stills or landscapes, because I could focus then release the shutter without the camera trying to refocus.
Sports Illustrated has a site for their photographers that defines the camera settings they recommend for pro-am and pro bodies. Their settings specify what custom settings need to be set to enable the split functions. The specs are a bit old, but should be good enough to get you there.