Complex sentence: The Asclepiads may have originally been members of a family claiming descent from the god of healing Asclepius, with the name only later being adopted by all doctors; or they may always have been an association of medical men venerating the god as their founder.
Split into simpler sentences: The Asclepiads may have originally been members of a family claiming descent from the god of healing Asclepius, with the name only later being adopted by all doctors. Or they may always have been an association of medical men venerating the god as their founder.