Slapping a compression bandage on a major wound requires no real skill and can buy you time, but unless you fix the major blood vessel through surgery (amputation or repair) the bleeding won't stop.
Don't sell your profession short and don't minimize what you do. I've been a firefighter for 19 years and have worked with dozens of EMTs and Paramedics and of course I've had training but not at the level they have. It's not as simple as slapping a compression bandage on. The First Aid skill in RM2 is stopping the bleeding with appropriate equipment and NOT resorting to a tourniquet and knowing which wounds to address.
Second Aid is suturing wounds (bleeding more than 5/rnd) and setting bones and can still be done in the field, but beyond that, if the wounds are that bad, far more skill is needed than battlefield triage.
Most of First Aid training is about not panicking in a critical situation
You are 100% correct. This is the First Aid skill at work. Not panicking, knowing which wounds to treat first and how. Knowing what not to move because the jagged bone could sever another blood vessel. Knowing to leave the arrow in the wound because it could be plugging a bleeder.
In game play, one modifier is -10 per bleed per round. First Aid can stop up to 5/round of bleeding... that means -50 to the First Aid skill roll. Without First Aid skill (or a low First Aid skill) just slapping a bandage on it, success is pretty low. In real life, and in gaming terms, I don't want anyone just "slapping a compression bandage" on me if their end result is -50 to their success! LOL With my luck, they'll apply pressure and make the blood squirt out faster from other holes.