I think the Scripture tells us several reasons why we do not see more healing in the church. Also, this seemed to be the case for the early church, and we should not be surprised to see the same situation today. For instance, sickness seems to be a normal situation in the early church just as it is today.
Consider:
1 Corinthians 11:29-30 KJV
29. For he that eateth and drinketh unworthily, eateth and drinketh damnation
to himself, not discerning the Lord's body.
30. For this cause many are weak and sickly among you, and many sleep.
There may be disagreement as to what "discerning the Lord's body" means, but the consequences are the same. MANY of them are weak, sickly, and MANY die young. This verse alone tells us that there were many people sick and dying in the early church, much like today.
Matthew 13:58 KJV
58. And he did not many mighty works there because of their unbelief.
Mark 6:5-6 KJV
5. And he could there do no mighty work, save that he laid his hands upon
a few sick folk, and healed them.
6. And he marvelled because of their unbelief.
And he went round about the villages, teaching.
This seems to say if people hold the minister in low regard it will affect the results of the ministering. Even Jesus experienced the effects of unbelief in His ministry. If some were not healed in His work, we should not be surprised if few are healed today. Today, people who preach and teach healing are often mocked and even called heretics. It is no wonder that there are few manifestations.
There is other evidence in the NT of believers being sick.
2 Timothy 4:20 KJV
20. Erastus abode at Corinth: but Trophimus have I left at Miletum sick.
Philippians 2:25-27 KJV
25. Yet I supposed it necessary to send to you Epaphroditus, my brother,
and companion in labour, and fellowsoldier, but your messenger, and he
that ministered to my wants.
26. For he longed after you all, and was full of heaviness, because that
ye had heard that he had been sick.
27. For indeed he was sick nigh unto death: but God had mercy on him;
and not on him only, but on me also, lest I should have sorrow upon sorrow.
We have no idea why these ones were sick. But, once again, we can conclude that despite the promise of healing and health in the NT, there were reasons why some were not healed. It was not unusual then, nor should we be surprised that it is normal for today.
James does not seem to think it an unusual thing for believers to be sick. He asks, "is any sick."
He states the prayer of faith shall save the sick. That is an unbreakable statement. If the prayer is offered with faith, the sick one will save the sick, and the Lord will raise him up.
James 5:13-15 KJV
13. Is any among you afflicted? let him pray. Is any merry? let him sing psalms.
14. Is any sick among you? 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:
15. And the prayer of faith shall save the sick, and the Lord shall raise him up; and
if he have committed sins, they shall be forgiven him.
The first point is that having believers sick and even dying in the church was not an unusual thing then, and we should not be surprised that it is also common today. Never did they state that the existence of sickness in the church was due to God or His will.
The second point is that never in the NT was the will or power of God called into question. The word was repeated over and over again. God did not and does not change. It may be in our procedure or our understanding. There is certainly a lot of unbelief in the church. Many even deny that healing is for today. Some doubt the minister. I have even heard people who come forth for healing say they did not know if they were going to be healed. That is called unbelief. The scripture says you will be and that "you have to believe you have received" "when you pray." Yet I am willing to bet the vast majority do not do this or even know they should. They wait to see what happens and then attempt to resolve the will of God by their failed prayers. "I was not healed. Therefore, God did not want me healed. Therefore, God does not heal all who come forth to be healed."
In the end, we need not question the veracity of God's will or Word today because of what we see around us. The Word is still true regardless of what they or we think or see.