So… I was an EMT at one point in my life.
Blood is good for it, you are correct. But it’s the body’s response and the fact that ot sends too much blood to the area that isn’t helpful, causing the fun things such as swelling, pain, and inflammation. This (elevating) actually promotes faster healing
Also, applying ice helps because it slows down the blood flow and swelling even more. If you watch any sports you’ll see that when someone is injured, almost any injury, the first thing the trainers do is apply massive amounts of ice. Keeping swelling to a minimum is extremely important for the initial moments of an injury. I now ice down pretty much anything that happens that hurts , and find that the consequent bruising, swelling and pain are greatly diminished and recovery is much faster. After handling the initial swelling, heat is then very beneficial to recovery. In most cases, early on after the initial icing and elevation, a cycle of ice followed by heat works very well. The ice slows down fluids and swelling, but leaves the area stiff and with little blood flow. The heat then loosens up the tissue and fluids, allowing blood to get in to help recovery. Best of both worlds.