Problem with glue is that it will only work after the game has finished or for very superficial abrasions. A decent cut will reopen as glue isn't strong enough.
For a decent cut to get back on you need to either stitch or staple it up. Stitching is difficult, needs practice and should really only be done by medicos or someone with similar medical training.
I actually wouldn't have a problem with team trainers using a staple gun to close wounds until the game finishes, then after the game send the wounds down to the local doc to have the staples taken out and be stitched properly. The problem with staples is they leave a much bigger scar than stitches and are more likely to get infected. Often I will staple during the game for speed and then exchange for stitches after the game.
I can't see how a team trainer stapling a wound up at a game where there is no doctor is any more dangerous than a player running around the field with blood streaming down his head or wearing a blood soaked turban.
The other thing you might need is to get a supply of guns (sterile single use only of course).
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