So the thing is, everyone is right. Sort of. Tourniquets were taught decades ago. Then they did fall out of favor. That was because doing something like garroting a leg or an arm with a shoestring tended to do a lot of damage to the vessels and nerves under the tourniquet, and also when they were left on for a long time the limb would likely be lost. Direct pressure, and to some degree pressure points, because the recommendation. More recently, tourniquets have come back in the context of military combat care. However, that is because it's combat care, and the idea is to keep people in the fight by letting them quickly apply a tourniquet one handed, and then fight on. That's spelled out quite clearly in TCCC guidelines. But those tourniquets are wide enough that they shouldn't do much immediate damage, except for pain. Then, it's usually possible to evac quickly to a medical person who can take the tourniquet off in a reasonable period of time, while maintaining control of bleeding. (Also, all this is being done in conjunction with dressings that have a blood clotting agent.) So, if you're in combat and get a limb blown off, put on a tourniquet. That concept has been spreading to civilian practice, probably quicker that it should, since in most cases direct pressure is still best. And that can mean fingertips pushing hard on a specific spot, rather than just a palm. Most places in the US, a store bought tourniquet probably wouldn't do too much damage before someone in a definitive care setting would be able to get it off. This is sort of akin to how backboards used to be the thing for possible spinal injuries, and now there's serious rethinking about whether that's actually good and necessary; thoughts change on medical care, and TKs are coming back.
I don't know whether this should be necessary diving equipment. Tourniquets have become something of a fad lately, with people buying them up for police officers to wear on their belts, and whatnot. They might occasionally have a use, but won't do much good without practice. Don't use them for every bleed, since most bleeds aren't all that bad, but if it's horrendous, go to it.
Although I don't know much about dealing with shark attacks, as I'm more of a hiker, I do know about avoiding bear attacks. For that purpose, I just carry a jar of bacon grease, find someone camping a ways off, and smear it on their tent. Keeps the bears away from mine. Maybe something similar would work with a dive buddy.
REMEMBER: All bleeding stops eventually.