Is church membership necessary?

The teaching of Reformed theology has always been that not every member of the visible church is a member of the invisible church. As you stated, a lot of people are church members but don't belong to the King. You also correctly stated that this is proven over and over again in the Old and New Testaments.

But, what about formal church membership? The Bible is our standard for determining truth, thus we must ask whether the Bible teaches about formal church membership, either directly or indirectly. While nothing in Scripture directly commands what might be termed "formal" church membership directing people of the body of Christ to submit to a particular local church, it is in the Bible indirectly by good and necessary consequence. As the Westminster Confession of Faith states:

The whole counsel of God concerning all things necessary for his own glory, man's salvation, faith and life, is either expressly set down in Scripture, or by good and necessary consequence may be deduced from Scripture ... and that there are some circumstances concerning the worship of God, and government of the church, common to human actions and societies, which are to be ordered by the light of nature, and Christian prudence, according to the general rules of the Word, which are always to be observed.
(WCF 1.6, Of the Holy Scripture).

What is this "good and necessary consequence?" Three things: the necessity of church order (1 Corinthians 14:40), church leadership - elders and deacons (1 Timothy 3), and the ministering of the gifts of the Spirit (1 Corinthians 12-14). These all speak not only to an organism (the body of Christ) but an organization (a visible church). One of the most important elements of formal church membership is submission to those who are in authority over you (Ephesians 5:21; Hebrews 13:17). If there are no formal church members, there can be no formal leaders. How then can there be submission when there is nothing to submit to? Church discipline (Matthew 18:15-18; Galatians 6:1-7) for the edification of the body of Christ would be impossible to do if not for formal church membership; there would be no set standards and no recognition of authority.

While the Bible speaks of "the church in Christ Jesus throughout all generations" (Ephesians 3:21), it also speaks of the church by locality and thereby implies local church membership.

Acts 11:22 News of this reached the ears of the church at Jerusalem, and they sent Barnabas to Antioch.

Acts 13:1 In the church at Antioch there were prophets and teachers: Barnabas, Simeon called Niger, Lucius of Cyrene, Manaen (who had been brought up with Herod the tetrarch) and Saul.

Romans 16:1 I commend to you our sister Phoebe, a servant of the church in Cenchrea.

1 Corinthians 1:2 To the church of God in Corinth, to those sanctified in Christ Jesus and called to be holy, together with all those everywhere who call on the name of our Lord Jesus Christ...

1 Corinthians 16:1 Now about the collection for God's people: Do what I told the Galatian churches to do.

The Bible doesn't only use the singular term "church," but also the plural "churches," implying a different churches at different locations, all unified under Christ.

Acts 14:23 Paul and Barnabas appointed elders for them in each church and, with prayer and fasting, committed them to the Lord, in whom they had put their trust.

Acts 15:41 He went through Syria and Cilicia, strengthening the churches.

Acts 16:5 So the churches were strengthened in the faith and grew daily in numbers.

Romans 16:16 Greet one another with a holy kiss. All the churches of Christ send greetings.

1 Corinthians 4:17 For this reason I am sending to you Timothy, my son whom I love, who is faithful in the Lord. He will remind you of my way of life in Christ Jesus, which agrees with what I teach everywhere in every church.

1 Corinthians 7:17 Nevertheless, each one should retain the place in life that the Lord assigned to him and to which God has called him. This is the rule I lay down in all the churches.

1 Corinthians 11:16 If anyone wants to be contentious about this, we have no other practice--nor do the churches of God.

There are also some passages that refer fairly undeniably to actual congregations (e.g., Rom. 16:5; 1 Cor. 16:19; Col. 4:16). We need to be thankful that God recognizes local church membership, otherwise his whole church would come under judgment (1 Peter 4:17). Even so, when we look at the churches at Ephesus, Smyrna, Pergamum, Thyatira, Sardis, Philadelphia, and Laodicea (Revelation 2-3) we see both localized blessings and cursings.

There are numerous texts that indirectly imply church membership. Paul's formal exclusion of the sinner at Corinth presupposes formal inclusion. Paul exhorts the Corinthian church to remove a brother from their ranks who was sinning in a way not even approved by pagans (1 Cor. 5:2, 7, 12-13). Paul's reference to "the majority" in 2 Corinthians 2:6-7 seems to refer to a group commonly recognized as the church's members. The early church kept a list of widows as mentioned in 1 Timothy 5:9. If widows were listed, it is likely that a list of members was kept and updated as well.

Here is a helpful quote from 9Mark Ministries:

By identifying ourselves with a particular church, we let the pastors and other members of that local church know that we intend to be committed in attendance, giving, prayer, and service. We increase others' expectations of us in these areas, and we make it known that we are the responsibility of this local church. We assure the church of our commitment to Christ in serving with them, and we call for their commitment to serve us in love and to encourage us in our discipleship. In short, we enter a covenant relationship with that church and its leadership.

The Bible teaches that Christ's authority cannot be "actually" diluted. He is God and all powerful. The Bible also teaches the concept of under-shepherding. Christ is called the Shepherd and Overseer of our souls (1 Peter 2:25), but individuals that Christ has appointed are also overseers (Acts 1:20; Philippians 1:1; 1 Timothy 3:1; 1 Timothy 3:2; Titus 1:7). From Hebrews 13:17, we see that the leaders or elders of the local church do have God-given authority to which the congregation is called to happily submit. Leaders exercise this authority for the provision, protection and the good of the congregation. Acts 20:28 speaks directly to this: "Keep watch over yourselves and all the flock of which the Holy Spirit has made you overseers. Be shepherds of the church of God, which he bought with his own blood."

More is said regarding this subject in Church Membership.