I have field dressings in my shooting kit , plus more in my car which is never generally far away when shooting. interestingly we did a risk assessment for range shooting at my club and a doctor we had in the club at the time said that there was a greater need for an AED/first aid training than field dressings because with the membership demographic a heart attack was more likely to need to be dealt with than a gunshot wound ?
as for edc , I have a grab bag in my car but on my person I generally have a knife , torch , lighter when outdoors . the grab bag is kept in a cubby hole in the boot of my car and is accordingly difficult to access , this should show the police that I do not intend to use it's contents for evil doings!