From Mark 16;
17And these signs shall follow them that believe; In my name shall they cast out devils; they shall speak with new tongues;
18They shall take up serpents; and if they drink any deadly thing, it shall not hurt them; they shall lay hands on the sick, and they shall recover.
19So then after the Lord had spoken unto them, he was received up into heaven, and sat on the right hand of God.
20And they went forth, and preached every where, the Lord working with them, and confirming the word with signs following. Amen.
If you read from verse 14 you will see this more clearly I think:
He is addressing the eleven remaining apostles:
14 Later Jesus appeared to the Eleven as they were eating; he rebuked them for their lack of faith and their stubborn refusal to believe those who had seen him after he had risen.
15 He said to them, "Go into all the world and preach the good news to all creation. 16Whoever believes and is baptized will be saved, but whoever does not believe will be condemned. 17And these signs will accompany those who believe: In my name they will drive out demons; they will speak in new tongues; 18they will pick up snakes with their hands; and when they drink deadly poison, it will not hurt them at all; they will place their hands on sick people, and they will get well."
"He rebuked them for their lack of faith and stubborn refusal to believe those who had seen him after he had risen"......then He told them what they should do and what they would see (possibly as proof and encouragement to them of the reality of truth of these people's conversions).
My point is He was talking specifically to the Eleven here about what they would see regarding those who believed and were saved from their preaching of the good news.
Do I pray for people to get healed? Sure. I've been praying for a friend to be healed of cancer. He died but when he was first diagnosed he was given a lot less time than what he lived he and his family are Christians and they went for chemo.
The bible does not forbid medical care.
I see the point you are trying to make but even if we take the part of your specific verse:
18...... they will place their hands on sick people, and they will get well."
It does not forbid receiving medical care.
Also,
Matthew 9:12 On hearing this, Jesus said, "It is not the healthy who need a doctor, but the sick.