All the sources I've read said that Naegleria fowleri enters the brain through the nose. But lets say that someone had cuts on their arm or leg and they are swimming in water that is conducive to Naegleria fowleri growth. Could it enter the cut into the bloodstream and then into the brain or cause some other type of damage if it enters the bloodstream?
No, Naegleria fowleri is a free-living excavate form of protist that lives in warm fresh water. Fowleri finds its way into the brain by eating through the olfactory neurons in the nose where it multiplies itself greatly by feeding on nerve tissue.
Once it penetrates the nervous tissue, fowleri's feeding results in significant necrosis of and hemorrhaging in the olfactory bulbs. The protist then climbs along nerve fibers through the floor of the cranium via the cribriform plate and into the brain.
Since the protist travels by feeding on nervous tissue, fowleri could not enter the brain via bloodstream.
Just some more facts about Naegleria: