I think this verse means to that we should pray for those people who are physically sick and in some ways I partially believe in a kind of limited cessation. You can see a very lengthy response to the state of all the gifts here Are the gifts still active?.
Although at the time this letter was written the healing may have commonly been extra ordinarily miraculous, however, just because we pray for someone that is sick does not mean that God will not answer by normal means such as bed rest, even at that time in history. Whether God answers miraculously, or just supporting the person under a time of temptation and giving him and his doctors wisdom to take the proper care, or whether God’s answer is to simply bring that person home into heaven the next day, of course prayer should be offered! So this verse does not relate directly to the question of cessation or not cessation. Whenever we are in distress we should seek prayer especially when we are truly sick.
Going to God in payer simply assumes that God could heal, or otherwise help us in our sickness. The answer to all of us, at some point in our prayers, will be comfort and strength to face a death we are peacefully prepared for.
Would we expect any Church not to encourage sick people to be prayed for?