When a person has serious wounds, it takes immediate attention to see that the most important issues are dealt with asap.
Lack of oxygen to various parts of the body is a condition that will keep getting worse.
Typically people with serious injuries will go into some sort of shock state.
A typical treatment for various types of shock appears to be epinephrine which is usually injected.
I suspect an induced coma would only be used if everything is known, under control, and carefully monitored.
They can always inject epinephrine if needed.
Without epinephrine at hand, having people "fight" for their lives seems a reasonable substitute.