The mechanics of focusing in the camera are the same, regardless of how you trigger the focus.
If you just keep the button pressed, AF-ON or half-press or whatever you use, then there's no difference at all, from a technical point of view.
The advantage I see with using AF-ON is that I can momentarily interrupt focusing, but still take pictures, if I see that something (bushes or whatever) are interfering with the focusing. If I'm following a runner who goes behind some bushes, I can just stop Servo AF from running while the runner is behind the bushes. Since the runner is in focus, but the bushes aren't, I can still take pictures that make sense, if I like. And I'm ready to continue as soon as there's a clear line of sight again.
There aren't many bushes on a football field, but there are other people, so the principle is the same.
Probably you aren't getting along with holding the camera in a way where you can easily operate the focus button and take pictures at the same time, so it's you who mess up subject tracking, and thus focus.
I don't use back button for certain subjects either. Around the house snapshots work better with everything on the shutter button.
When shooting action I have both AF-ON and * active for focusing, one with the selected AF point, the other with the registered point (HP, Home Point). Like in the situation below, where I used a focus point a bit to the right in the viewfinder for photos of cars going to the left, but a point a bit to the left when they've turned around in the hairpin and moved to the right again. Just moving my thumb between AF-ON and * made the selection easy.
Should I need AE lock, I have that on M-Fn.
The heavily reduced size of the image below makes both cars look like they are out of focus, but the one I aimed at actually is sharp. It's better here.
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