Was Dr. Livesey's treatment a valid one in the Treasure Islands?
The procedure described here is called venesection. It is no longer standard of care, but was when Treasure Island was written.

Here is an excerpt from the first edition of Cecil's Textbook of Medicine (1927):


  During the acute apoplectic attack the following therapy may be employed: The patient should be put to bed with the head elevated. If he has fallen to the ground the clothing around the neck must be loosened. Difficulty in swallowing contraindicates the administration of nourishment by mouth. An enema may be given. When the face is puffy, the pulse full and strong, and there is reason to suspect hemorrhage, venesection is advisable. It should be employed only for plethoric individuals, not for patients who seem asthenic or arouse suspicion of thrombosis. Usually 250 to 300 cc. of blood are removed.