TC3, baby.
Strap a tourniquet on, fight the fight, and get the heck off the "x". Compromised airway? Dude's probably dead anyway. Heart stopped? Dude is day-ud, son.
And that is basically what CLS boils down to, as far as practical application. Sure, there is a bunch more to it, once you are in a safe area. But step one for bleeding from an appendage is to slap a tourniquet on the wounded limb, regardless of flow rate. Step 2 is haul a$$.
The scope of practice of a CLS is significantly greater than the scope of practice for someone with a "First Aid" / basic CPR. The scope of practice for a CLS is generally smaller than it would take to be truly useful outside of a firefight, or the immediate aftermath of a similar situation. If your scope of practice is <= a CLS (joking referred to as a Combat LifeTaker, because of how often we screw things up), you probably shouldn't be volunteering to go rooting around in bleeding people.
"Medic" is a title that is reserved for the folks who can and will snatch the life back into somebody with their bare hands in the middle of the shit, not somebody who knows which side of a band-aid has the sticky stuff on it.
Just saying, is all.