Since you're only after different solutions to the bleed out rulings, I hope unpublished systems are ok.
As I made my own system, I felt I wanted it to simulate at least the sense of being realistic. But I also hate bookkeeping, so naturally I wanted to design something that let minor things be ignored and only the major stuff be recorded.
So there are 4 types of injuries: Scratches, Light wounds, Severe Wounds and Critical.
Only Critical injuries is something you might bleed out from.
If in the head, Dead (or coma for 3 months)
If in any of the extremities (arms, legs): Incapacitated and Bleed out in 1d6 + Con bonus minutes.
If in the body, Incapacitated and halve time for bleed out.
For every extra Critical injury, subtract 1 minute for each.
Saving an injured then requires two rolls. One to slow down the bleeding, one to stabilize the injury.
Medic/First Aid replaces the 1d6+con roll, but halving for body hit and extra criticals still apply. So it doesnt stop the bleeding, but it does give the character more time to get proper care. There is no difficulty to ovecome at this roll, as the result simply replaces the natural bleed out duration. Once that time is up, you can't reroll for "more time" unless you have access to blood transfusion in some way. (donor, plasma)
Medic/Surgery stabilizes the injured if a certain difficulty threshold is reached, and that difficulty increases with every extra non-fatal wound. Since each stabilizing attempt takes a minute, and can be rerolled until the patient stabilizes or dies, it does give characters a pretty good chance to survive. Healing time however, is a totally different matter.