I have given blood many times, have dealt with cuts and bloody noses, have shot and field dressed animals when hunting, etc. Never a problem. When someone else is injured and bleeding I actually get a bit of adrenaline rush and feel like I can think clearly and rationally in an emergency.
When I was in college I was so confident that these things didn't affect me that I volunteered to let a friend who was training as an EMT practice inserting a picc line into my arm vein. No issue when she inserted the catheter into a vein on my hand, but as I saw and felt the line moving up inside the vein I felt a strange sensation as the blood seemed to rush out of my head . . . my knees buckled and someone had to catch me and guide me to a sitting position on a nearby bench.
It was just for a moment, and I never lost consciousness, and it was a completely involuntary thing that happened. But I was immediately embarrassed that it happened, even though I knew I had nothing to be embarrassed about. In my mind I had (and probably still do to some extent) associated fainting with weakness, the inability to deal with stress or something like that. But I think sometimes its just a physical response that has little to do with the person's own tolerance of stress or emergency situations — sometimes it just happens and there is nothing the person could have done about it.