Thank you for your question to us about the significance of membership. I'll first point you to a small booklet that we publish that might be useful, and then offer a few helps. The booklet is by Mark Brown and Larry Wilson, called "Why Join a Church?" You will find it on the Publications page of OPC.org.
Regarding specific references, I would point to Acts 2:42–47. Note especially the last verse; being saved and being joined to the local church are distinguished, but not separated. All those the Lord was saving were being added to the numbers of the local church. And note also that the LORD was the one both saving and adding; it wasn't a device contrived by men.
You might also reference I Corinthians 11:17ff. regarding the Lord's Supper. Paul's rebuke is based on the fact that the table not only declares the union of the believers with Christ, but also with each other. Their selfish approach to the needs of their brethren meant that Paul regarded their sacrament as not being truly the Lord's Supper (v. 20). There was a mutual connection among the believers that was to be demonstrated at the Lord's Table. Historically, only those marked by God's people as belonging to God's people could take the Passover (Ex. 12:43ff.). So God has always separated those united with him and with each other from the world.
Finally, a comment about the wording of your family member that membership is "man's way of 'excluding the public from worshipping God.' " I cannot make sense of such a statement. I've never known a church that tried to prevent people from coming to the worship service. Faithful preaching of the Word calls men to worship God; that call is proclaimed to everyone within earshot! It appears that the family member is saying that the only real worship is taking place at the Lord's Table. That's not what the Bible says, so he/she needs to be corrected on that. Worship is an activity of the heart; the activities are to reflect the heart as God directs those activities by his Word. And his Word has always separated those who are part of His people from those who are not.
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