The article references James 5:14-15 as a common justification for prayer-without-medicine: "Is anyone among you sick? Let him call for the elders of the church, and let them pray over him, anointing him with oil in the name of the Lord. And the prayer of faith will save the one who is sick, and the Lord will raise him up. And if he has committed sins, he will be forgiven." I fail to see how that passage of Scripture prohibits use of medicine. In actuality, not even medicine truly heals the sick, as any doctor can tell you... the medicine simply helps your body to heal itself. Pray, believe God, and then if you need it, take some medicine. Get a procedure done, get surgery, take a pill or a shot. My pastor just got knee replacement surgery. You can bet that he has been praying, his wife has been praying, and his whole church has been praying for him. It is not a lapse of faith to allow clever medical practicioners to do what they do best in addition to praying in faith. You may just find that after your procedure, God "raises you up" and you recover so quickly that even your doctor is surprised!
The thing that actually came to mind, though, was a parable Jesus told, recorded in Luke 10:25-37. It's a very familiar story... a man gets robbed and beat up, and two religious people see him but refuse to help him. A non-religious person comes along and helps the man, and is commended by Jesus. I don't think there is anyone who would argue that the "Good Samaritan" did anything sinful or wrong in helping the man; again, Jesus basically commended him for his actions. What did he do that was so great?
[The Samaritan] came to where [the injured man] was, and when he saw him, he had compassion. He went to him and bound up his wounds, pouring on oil and wine. Then he set him on his own animal and brought him to an inn and took care of him.The Samaritan was commended for administering first aid! He gave the man medical attention! He didn't "pray in faith" and then put the man in the hotel to wait for his miraculous healing... he "poured on oil and wine" (first aid with medicine) and then "took care of him" (additional medical attention).
[Jesus asked the man He was telling the story to,] "Which of these three, do you think, proved to be a neighbor to the man who fell among the robbers?" He said, "The one who showed him mercy." And Jesus said to him, "You go, and do likewise."