I agree with you Terry. I have also done my own research on that subject. For instance, Papias said that Matthew wrote in Hebrew language a collection of sayings. This seemed to be the Gospel used by Nazarenes and Hebionites, who were christians that rejected the divinity of Jesus, regarding him only as the Messiah. Nevertheless, Jerome, an "orthodox" church father at the end of the fourth century wrote that most christians believed that that Gospel used by these sects was the original one written by Matthew. And if you read Jerome's and Eusebius' quotations from that Gospel you will notice that many statements differ from that of the greek version of Matthew's Gospel. So it means that many christians of the fourth century were not convinced of the authenticity of greek version of the Matthew's Gospel that we currently have.
But we have only to read the current version of Matthew's Gospel to know that Matthew could not write it. For instance, we read in Matthew the following:
Therefore go and make disciples of all nations, baptizing them in the name of the Father and of the Son and of the Holy Spirit (Matthew 28:19)
Jesus could not command that, because if it were true, gentiles would have been baptized as early as Jesus died, and we know that it happened after some inner debates in the Church. And we read from the book of Acts that Peter was primarily reticent for baptizing Cornelius. So it shows that the author of Matthew 28:19 could not be a direct disciple of Jesus. This is also an example of the many words that Jesus did not say which appear in the Gospels.