Paul claims to have certain rights as an apostle, just as Peter or the Lord’s brothers do. And he agrees that the church ought to support those who serve in the ministry. But Paul does not insist on those rights because it would be a hindrance to the gospel. Paul does not want anything to become a wall keeping people away from hearing the gospel.
The Corinthian church seems to have support other apostles, maybe Peter visited there, although it is unlikely James ever did. If Peter showed up in Corinth with his wife (and a small entourage, the “support staff”), then the church gave him a place to stay and made sure he and his wife had all their needs met while they are in town. If Peter is due that kind of respect, Paul says, how much more is Paul due that kind of respect? He was the apostle who first brought the gospel to Corinth and labored in their church the longest. Paul is an apostle with a right to have the Corinthian church take care of his needs (food and daily needs, a place to live, and other expenses).
But he has not made use of his rights so he will not be an obstacle in the way of the gospel.
Paul has rights as an apostle, but also as a Roman citizen. In Acts 16, he insisted on his rights as a citizen after he was arrested, beaten, and imprisoned without due process. But when Paul was arrested and stood before Gallio in Acts 18, he does not mention his citizenship. However, that he was given due process of the Law and received justice from the Roman governor himself may imply he was known as a Roman citizen and received his legal rights. If Paul had access to some wealth, he has more rights in Roman society since he would be seen as Roman citizen of means who can travel extensively through the empire
He sets aside rights as an apostle so that he does not become a hindrance for the preaching of the Gospel. An obstacle (ἐγκοπή, only here in the NT) is something that causes a hindrance, since as great rocks and streams which are obstacles for travel through difficult territory (πολλάκις διὰ τὰς ἐγκοπὰς, Diodorus Siculus Hist. 1.32.8).
Were there people in the early church who sought to make a profit from the preaching of the gospel? An ancient equivalent of a modern televangelist?
Some philosophers were known for charging exorbitant rates for for their services. Philostratus rebuked a Stoic philosopher Euphrates for “money-grubbing and selling wisdom at a discount” (Life, 1.12.3). Paul will describe the so-called “super-apostles” in 2 Corinthians, teachers who do in fact insist on their right to support (and apparently good support at that!)
Didache 12, Beware the Christmonger! The noun χριστέμπορος only appears in Didache 12:5, combining Χριστός and ἔμπορος “trader.” BDAG suggests “one who carries on a cheap trade in (the teachings of) Christ;” BrillDAG, “trafficker in Christ.”
Didache 12 But everyone who comes in the name of the Lord, let him be welcomed. And then, having tested him you will know him. ⌊Then you will be able to distinguish whether he is true or false⌋. 2 If the one coming is a traveler help him as much as you are able. But he shall remain not among you ⌊more than⌋ two or three days, if he has need. 3 And if he desires to stay with you, being an artisan, let him work and let him earn his keep. 4 And if he has no craft, take this into consideration according to your understanding how he shall live among you as a Christian without being idle. 5 And if he does not want to act in this way, he is a Christmonger. Beware of such as these. (Rick Brannan, trans., The Apostolic Fathers in English)
Paul does not want to be known as a Sophist or philosopher gathering wealthy students to himself to become rich and powerful. Nor does he want his relationship with the Corinthians to resemble a patron/client relationship. He denies anything he might have by rights if it is a hindrance to the Gospel.
Paul did not insist on this right when he was in Corinth and he not demanding it now (9:15-18). Why? If he is paid to preach the gospel, he has “no ground for boasting.” It is a job requirement! If Paul is just doing his job, he does not have the right to boast (9:15-17). In fact, Paul would rather die than be deprived of his grounds for boasting (9:15) By preaching the gospel in Corinth without demanding his rights as an apostle or Roman citizen, Paul can boast in what God has done and he will receive his reward for being faithful to his commission.
This is challenging to modern the modern Church because we have created a professional class of Christian ministers, many expect to be paid a decent wage befitting their education and preparation. Some have taken additional education in counseling and other people-oriented skills in order to be better prepared for helping people i local churches. I do not think well-trained ministers are the problem, and churches should not expect their pastors to live in poverty or take two or three side-gigs just to make ends meet. But there are many pastors in this world, in America or the rest of the world, who are clearly in it for the money. Sinclair Lewis’s Elmer Gantry is almost a hundred years old now (or watch the 1960 movie), but the problem of pastors and evangelists demanding more money from their congregations is still one of the biggest hindrances to the gospel.