There is no doubt that knowing basic first aid and perhaps even more in depth medical training would be beneficial in a bad situation.
However, it is a perishable skill that you need to keep up on and that is just as important as doing it in first place. Its not the hardest thing in the world to remember when you have a few minutes to think about it, but you most likely won't have a few minutes to do so in a dire situation. You can lose consciousness in about 30 seconds from a femoral artery bleed. So with that said, you must also be able to apply first aid to yourself. We did a pretty extensive medical training exercise, and applying a tourniquet one handed in a timely fashion can be quite difficult. Am I saying everyone needs to be a Corpsman or Medic? No. Though, most CCW holders feel it's better to have a gun and not need it, than not have it and need it. Why not the same for lifesaving skills? To be honest, with all the dangers in the world that don't concern a gun, not having those skills can be more dangerous for your family or yourself than situations involving self defense, IMO.
"In 1968 for my senior field trip I was sent to RVN"
"It is far more important to be able to hit the target than it is to haggle over who makes a weapon or who pulls a trigger." -Dwight D. Eisenhower