What weight should be given to the preferences of non-Christians for what Christians do in their gatherings?

There are four biblical principles to keep in mind here.

1. God alone determines how he is to be worshiped (Lev. 10:1-3; Ex. 20:2-6; John 4:20-26). In his Word, God tells us that regular Christian gatherings should contain singing (Eph. 5:18-19), Scripture readings (1 Tim. 4:13), prayer (1 Tim. 2:8), preaching (2 Tim. 4:1-2), baptism and the Lord’s Supper (1 Cor. 11), and not much else. Therefore, Christians are not free to add or subtract elements of public worship services according to their preferences, non-Christians’ preferences, or anyone else’s preferences.
2. Everything in public gatherings of Christians should aim to edify Christians (1 Cor. 14:12, 26). The primary, overarching concern governing Christian gatherings should be how each element contributes to Christian growth.
3. We should labor to put no offense in front of the gospel (1 Cor. 9:1-23). This doesn’t mean that we get rid of preaching if our non-Christian friends don’t want to hear preaching. It does mean we should strip away any unnecessary, distracting, and potentially offensive cultural practices that can get in the way of the gospel.
4. Our services should be evangelistic and intelligible to non-Christians (1 Cor. 14:24-25). Even though building up believers should be primary, we should also be concerned to communicate intelligibly to non-Christians who come to church. This means that while non-Christians’ preferences shouldn’t dictate what is done or how it’s done, we should be sensitive to them so that our services are more intelligible, and therefore more evangelistic.