I have seen patients saved by the use of an improvised tourniquet in some regional retrievals, but it is uncommon (and usually involved heavy machinary or large water based man eating creatures). Maybe I am missing the inside scoop on those that never made it to hospital because of the failure to provide or improvise a tourniquet in the time frame required. I personally wouldn't carry a specific tourniquet, but instead ensure I have at least 1 thing that is immediately available to improvise with. My go to would be a sturdy belt and a sturdy clamp to maintain tension. Whatever you use (or plan to use) should be a minimum of two inches wide to generate adequate deep pressure. I do not personally see the benefit to carrying a commercial tourniquet on my person because it serves only a single function, for an uncommon injury.
Having said that, I only really carry the appropriate clamps in my car based FAK/trauma kit, where the addition of a tourniquet wouldn't really make a weight or space issue anyway. The reason I carry a more comprehensive trauma FAK in my car is that I am more likely to come across the need to use it by road. In the scenario of a road traffic accident with mangled limbs, car seatbelts make excellent tourniquet material and, along with the trusty clamp, would be my go to over my belt (seeing as I don't carry commercial tourniquets). Maybe I will revisit the inclusion of at least one appropriate clamp in my pack for any journey on foot.
On another note, I would only ever recommend using a tourniquet (improvised or otherwise) if you are adequately trained in there use. If they are inappropriately applied (i.e. below the arterial blood pressure) they will in fact not stem arterial bleeding and make venous bleeding worse due to impeding venous drainage. High and tight, if it doesn't hurt like hell its not on tight enough. Be aware that bleeding may come on later as the persons condition improves due to a higher blood pressure that might overcome the previously adequate pressure exerted by the tourniquet.
From what I have seen, many Americans do seem to love carrying tourniquets in their FAK, but then they also frequently have a handgun in their daily carry...