Yes, you are 100% correct. They were chosen instruments through which God chose to publicly work. Perhaps I am guilty of using a little shorthand. :)
It was always the faith of the "heal-ee" not the acumen of the "heal-or" that caused and causes healing.. spiritually.. by the Holy Spirit or Jesus.. Could be the Holy Spirit Could Not/CANNOT heal in certain "venues" because it would link him to the practises "held/performed/sinned" there..
There is many kinds of healing not all physical.. even an epiphany from the Holy Spirit is a kind of healing.. Call it a mental healing.. Example: Even knowing what a messiah is and seeing that Jesus is "that one" is a healing.. Jesus is not MERELY a Super-Hero with a fleshly uniform he much much more.. Actually Jesus' flesh is a metaphor of who and what he really is.. "Jesus" existed long before he put on flesh.. and still does.. Jesus is a Spirit..