Sam, who says that a person must have that intent in order to be cured? It certainly is not necessary that the doctor thinks that as we even listen to idolatrous doctors (SA OC, 618:1). The Taz says (YD 336:1) that no one is on that level. The Netziv says (Heemek Devar Bamidbar 6:11 and 17:3) says that a person who acts in a manner inappropriate to his level is called a sinner. In fact, several Amoraim went to bloodletters. See also Rambam in his commentary on Pesachim Ch. 4 d”h shisha devarim asa Hizkiyahu.
As for the O.P.’s question, the shot is needed as a booster. It increases the chances of not getting it again. It is also not certain how long the recovered person’s immunity lasts.