Two of the biggest problems are bulk and weight.
If you load a bag up with all the stuff you would like to have it often ends up not being EDC or even car cargo because it is too bulky and heavy.
If it is not with you it is unavailable to you when you need it and then it is no good at all.

Bandages can be too easily improvised at the scene from clothing or other stuff to sacrifice a lot of space to carrying them. Field expedient bandaging might not be very sterile, but once the person you have done first aid on gets to the trauma unit the doctors can fill them full of whatever antibiotics they please.

That is why you see a lot of first aid kits that are little more than a few bandaid strips, adhesive tape, gloves and possibly a valved resuscitation mask.

The biggest part of your first aid kit should be what you carry between your ears.

edit:
Make sure you include a good watch and note taking material.