Yes, God has healed people, and will continue to do so, but it seems clear God does not let His son's name be wielded as a magic wand.
I'm not sure what you mean by that, as the Bible clearly states that we are healed through faith in the name of Jesus.
Mark 16:17-18: "And these signs shall follow them that BELIEVE, IN MY NAME shall they cast out devils...they shall lay hands on the sick, and they shall recover."
1 Corinthians 12:4-11 (NIV):
There are different kinds of gifts, but the same Spirit. There are different kinds of service, but the same Lord. There are different kinds of working, but the same God works all of them in all men.
Now to each one the manifestation of the Spirit is given for the common good. To one there is given through the Spirit the message of wisdom, to another the message of knowledge by means of the same Spirit, to another faith by the same Spirit, to another gifts of healing by that one Spirit, to another miraculous powers, to another prophecy, to another distinguishing between spirits, to another speaking in different kinds of tongues, and to still another the interpretation of tongues. All these are the work of one and the same Spirit, and he gives them to each one, just as he determines.
What does the "gift of healing" mean here? Clearly Paul was not under the impression that all believers had it.