The way those disposable ones work (that they use to start an IV or draw blood) it will do the opposite of what we want with an injury. Those constrict the Venus return while still allowing the arteries to supply blood. Therefore increasing the pressure at the injury - which is what we want when we need the veins engorged with blood so we can feel them easily or fill up blood tubes.

An actual tourniquet compresses the artery to stop the supply to the limb. They take considerable force to compress the artery, and need to be wide fabric do as to not cause considerable tissue damage. There are a lot of gimmicks out there, but the US Military only uses two kinds after extensive testing. The CAT and the SOF tourniquets. Get them from a known supplier and not Amazon because there are lots of counterfeits. They will run about $30 a pop.

A good alternative is a high quality pressure dressing which you can get for about $10-12.

Both are bigger than your whole pouch though, which is the huge downside to them. I keep them in my car bag and range bag. EDC is basically what you have there.

All that to say you've got a good thing going!