We do agree. Matthew's gospel is inspired, thus authoritative. During the 1st Century our evidence is Matthew himself and the other apostles. Most scholars put Matthews gospel as completed and distributed around 62-65 AD. (http://www.errantskeptics.org/DatingNT-ChronologicalOrder.htm)
Then if we bridge this with what the ECFs said we see the continuity. Here is what Ireneus had to say:
Irenaeus, around the year 180 gave the first historically documented list of the four gospels and their authors:
"Matthew also published a gospel in writing among the Hebrews in their own language, while Peter & Paul were preaching the gospel and founding the church in Rome. But after their death, Mark, the disciple & interpreter of Peter, also transmitted to us in writing what Peter used to preach. And Luke, Paul's associate, also set down in a book the gospel that Paul used to preach. Later, John, the Lord's disciple --- the one who lay on his lap --- also set out the gospel while living at Ephesus in Asia Minor. (Against Heresies 3.1.1)"
So you’re saying all the evidence we have that Matthew wrote a gospel and that his gospel is inspired, is based, entirely, 100%, upon Church teaching rather than upon anything whatsoever in scripture?
I agree.