incognito,
you have either cleaned many a tuna or you found a japanese tuna cleaning book. your description was right on, not many western fisherman bleed from the pec fin. they usually bleed from the tail which works well but if the fish is not spiked correctly it will shake the tail off making it difficult to boom the fish into the hold. after making the cut at the pec fin try making a small incesion under the gill plate (nape), just large enough to insert a deck hose. the pressure entering the body cavity will cause the blood to squirt 5 feet into the air. hold pressure on the outside of the gill plate until the blood runs clear. believe it or not, sometimes it helps to pump the pec fin to get the blood flowing.
tuna caught the same day will be tough, let the tuna rest until the meat is throughly chilled in ice (a brine works well). the loins can be cut and even made into sashimi the first day, after refrigerating overnight the meat will become tender.