Of all our Lord's miracles only one is recorded in all four Gospels. That unique position would suggest that all four writers considered this incident to be a central part of our Lord's ministry.
The feeding of the 5,000 appears in all the Gospels because its message characterises the primary focus of Jesus' teaching. The Pharisees and "the righteous" had an obsession with their ritual purity, especially at meals. They washed their hands before meals, and were particular about with whom they ate. Yet here, in this remote place, Jesus tells a group of more than 5,000 people to sit down and eat with Him. There was no possibility of washing hands, and certainly no way of checking if there were any sinners, tax collectors, or "unrighteous" in the crowd. The Lord dramatically and miraculously breaks down these religious barriers between people and invites them to eat with Him at His table. No one was excluded.
The early Christians took up this practice and "they devoted themselves to the apostles' teaching and to the fellowship, to the breaking of bread and to prayer ... They broke bread in their homes and ate together with glad and sincere hearts" (Acts 2:42, 46). Their meetings were based around a shared meal (called "love feasts" in Jude 12) - commonly called "breaking bread". This term is used elsewhere to describe sharing a meal together, any meal, and is not a unique descriptor for the Christian ritual called Communion, Eucharist, the Lord's supper, or Breaking of Bread. The meal probably began with the host saying a blessing over bread (kiddush), and ended with a cup of wine after supper (1 Cor 11:25). From beginning to end it was "in memory" of Jesus and was a re-enactment of His open table fellowship.
There is not a hint of a suggestion anywhere in the New Testament that children or visitors were excluded from any part of the meal. No where does the Bible say that this Christian ritual is for baptised "members only", or that people were excluded from it because of some sinfulness on their part.
Luke records a remarkable incident when Paul "broke bread" with his fellow-survivors of a ship wreck. After the storm "he took some bread and gave thanks to God in front of them all. Then he broke it and began to eat. They were all encouraged and ate some food themselves." (Acts 27:35-36). These words are almost identical to the way Luke records the last supper: "he took bread, gave thanks and broke it, and gave it to them" (Luke 22:19) and an incident when Jesus broke bread with some disciples after His resurrection: "When he was at the table with them, he took bread, gave thanks, broke it and began to give it to them" (24:30). In fact, Luke highlights that it was through this breaking of bread that these disciples recognised Jesus (v. 35) - no doubt because table-fellowship was such a central part of our Lord's ministry. Notice how each record has these words: (1) took bread (2) gave thanks (3) broke it and (4) gave it to them. (See also 1 Cor 11:23-24 where the same words are carefully recorded as the foundation for the celebration of Communion by the church). The breaking of bread after the shipwreck followed the same form as the breaking of bread ritual of the believers.
But we should also note that Paul "broke bread" with people who were strangers to him, "sinners" and non-Jews alike. They had been through a common experience and been saved. They were invited to the Lord's table in celebration. Perhaps we need to re-think what it is we celebrate at the Lord's table, and with whom.