Is there a biblical precedent for a church leader asking a church member to write and publicly read before the entire church a letter confessing a personal, private sin—such as impurity or materialism?” —A. O.
I am unaware of any such biblical requirement, nor does one find this practice in the course of church history. We have never had such a custom in any of the dozen or so congregations we've been part of in the last 40 years, though at times wandering sheep have made a short statement to the body on returning. I suppose that's different to what you're asking about, since the wanderers have been away for some time, and clarification for the body may be useful (James 5; Luke 15).
James 5:16 speaks of one-another confession. It's a biblical principle too often neglected. Private confession seems to be in view, for instance to one other Christian, or perhaps to the church leaders who will be setting aside some time to pray for the individual. Even then care must be taken, as Paul reminds us in Gal 6.
Of course if the sin in question directly involves multiple church members, then it will have to be addressed publicly—and I would defer to local leadership to know how best to do this. In 1 Cor 5 Paul had to address a messy sexual situation. He did not refer to the parties by name, but apparently the Corinthians were well familiar with the debacle.
As for a letter, through most of the history of God's people -- under both covenants -- the average member wasn't necessarily even literate. The only reading of letters in NT times seems to have been when an apostle's letter arrived (Col 4.16; 1 Thess 5:27).
Confession before the body might be helpful in some cases, for instance if the brother or sister strongly desires to make a statement, but there is no scriptural mandate. We do see a group confession in Acts 19, where the believers come clean of the sin of sorcery. Yet it seems spontaneous, rather than a requirement imposed by church leadership.
In the letter-reading scenario you describe, I'd imagine most members would be humiliated -- not humbled -- and the shaming process could easily frighten others into covering over their sins lest they too be publicly shamed.
Different churches will follow different practices, and embody different cultures. That isn't necessarily a bad thing. Yet these various responses to the teachings of scripture must be shaped by sound theology. God's way does not give carte blanche to follow or impose customs out of sync with the gospel. Any practice specified by scripture may be wholesome and fruitful, when it is applied consistently and without prejudice. As we explore this question, we need to compare our thinking and traditions with what is clearly stated in the New Testament. If it's unclear, or if our practice lacks biblical warrant, we should be willing to modify (or jettison) it accordingly.
Church leaders are tasked by God to lead by example. Do they read letters if they themselves lapse into laziness or materialism? Do they model lives of gentleness and respect, or authoritarianism? (My podcast on Lording it Over Others may be especially relevant.) Are the overseers actively involved in strengthening those who stumble? When the shepherds have the heart of Christ, they will lead firmly, lovingly, humbly, and personally.
Does this help? There are also various articles on confession at the website. I hope I haven't been unfair or misread between the lines. My desire is for congregational health, spiritual maturity, righteous leadership, and biblical fidelity.