I'm of the same mind set - first aid, in my opinion, isn't a boo-boo kit for splinters and minor burns, or coping with constipation. That's second aid.
First aid is keeping someone from bleeding out, keeping them breathing and holding the line against further trauma. So that's why, when I'm out and about in NYC, the traditional FAK stays in the computer bag, but on my person is a TAC-PACK like kit

1. Bleed-X
2. Tegaderm Transparent Dressing
1. Pair Purple Nitrile Gloves
1. 6x6 Sponge
2. 4x4 Sponge
1. Roll 2" Guaze (flattened)
1. Roll Flat Duct Tape (better than medical tape)
1. N95 Mask
1.. Vionex Hand Wipe (not carried on airplanes)
24" 550 cord
1 medic shears.

If you can't stop the bleeding, pack stuff in the puncture, tie off the amputation, all the benedryl in the world won't matter. My kit fits in a tiny little pouch, is airline-proof (yes, you can bring medic shears on board) and bleed-x is a powder.