Why can't the body successfully heal an Achilles tendon injury?

Needs rest. If you continue to walk on an injured area it does not get rest to help the body heal it.
Not if fullly torn. If the achilles tendon is inflamed it can be treated with rest, nsaids and pt. If is a slight partial, you may just need to cast or protect it for a while. But if the tendon is totally torn, it will likely need to be repaired because otherwise it will not heal with normal strength necessary for sports and have a higher risk of retear.