They were chosen before the Sermon on the Mount was delivered , but Matthew did not mention it in connection with that sermon. Hence it is not fair to say that Matthew knows only the charge to heal the sick , important as that is. The physical distress was great, but the spiritual even greater.
Power is more likely the idea of exousia here.
Today we have hospitals and skilled physicians and nurses, but we should not deny the power of God to bless all these agencies and to cure disease as he wills. Jesus is still the master of soul and body. But intelligent faith does not justify us in abstaining from the help of the physician who must not be confounded with the quack and the charlatan. The names are given here, but Matthew does not say that they were chosen at this time. The apostles disputed a number of times as to which was greatest.