Interesting questions. It could be his gift was evangelism and not 'follow-up'. Others thing it was important for the apostles to be involved in the conversion of the first foreigners post Pentecost.
Your memory is a little off on this one. James wrote to the church about calling for the elders. The next part of the chapter says to confess your faults one to another and pray one for another that ye may be healed. So not elders could minister in healing, but also all members of the body of Christ.
You are treating one verse as if it contradicts or overrides another. The Bible shows many ways people can be healed. One is through Jesus or the apostles laying hands on them. Another is for 'them that believe' the apostles message to lay hands on the sick. Another is through the elders anointing the sick with oil and praying the prayer of faith. Another is through confessing sins to one another and praying through one another. Psalm 107:20 says, "He sent his word, and healed them, and delivered them from their destructions. " People can be healed by the word of God. And plenty of people are healed because God is merciful to us. Unbelievers get cuts and colds that heal up.
A reference to one means of healing does not negate other references to healing. And James could be an early epistle.
God can heal supernaturally, but Paul still gave Timothy health advice. Paul himself had an infirmity that led him to first visit the Galatians, which probably occurred around 14, but clearly by Acts 16 at the latest, but performed great miracles after this.
Have you actually read these documents or studied early church leaders writings? The Didache and Justin Martyr refer to prophets among the Christians at that time, and the Didache gives detailed instructions. I'm not sure if they mentions healing. Ireneaus, a missionary to Gaul, had known Polycarp when he was young and Polycarp was old. Ireneaus wrote of brethren healing, miracles, prophesying, having foreknowledge, speaking in tongues, etc. He considered rejecting prophecy to be a characteristic of the heresies, and also thought that all churches should be able to raise the dead as his had.
Tertullian wrote of healing and casting out demons in his day. This has some quotes:
https://www1.cbn.com/signs-and-wonders-early-post-apostolic-era
If you read I Corinthians, it is implied that the Corinthians were violating the order that Paul was instructing them about, yet their gifts were genuine. In Seymour's newsletter, at one point, he wrote about the need for order when it came to speaking in tongues, and having interpretation of tongues. If they had genuine gifts and used them in a disorderly manner at first, like the Corinthians were doing, that doesn't mean the gifts were false. The Corinthians' gifts were not fake. And I have come across three or four references to people speaking in tongues that people knew at the Azusa Street revival, and the interpretation of a message in Russian was confirmed by a Russian speaker. One of the attendees said people hearing their own language is part of what drew the crowds.