IMHO there isn't a whole lot of point to individual players carrying first aid gear at most events; as the vast majority of games will have players within a 5 minute run of the staging area. However, when taking part in MILSIM type games that can range over 50+ acres, it's not a bad idea to have some people carrying the basics.
In that scenario, most teams/groups worth their salt will have basic medic gear with them to deal with the most common sorts of injuries, ie accidental falls. So gear to deal with sprained ankles, broken limbs etc are your best bet. You can toss a splint in the back of your assault pack and you'll never even notice it until you do need it.
As someone who has been severely injured during a game (fall from a watch tower, 2x fractured ribs, lacerated kidney) what helped me wasn't anyone's medical gear, but having some trained EMT's who were playing, and a good friend's willingness to drive a jeep to me and back out of the playing area to meet the ambulance (no matter how much I screamed like a bitch).
In short, if you want to carry some medical supplies, go ahead, just don't go overboard w/ portable defibrillators etc