Addition 180204: I regard this theory as improbable – I just need the theory to be formulated
There are a lot of theories of how the synoptic Gospels (Mark, Matthew, and Luke) were written and how they relate to each other. The academics proposing those theories are making a lot of text analyses, comparing Greek language with the language of certain centuries in the Antiquities. I have no competence to contest anything of that, but I can make a simple skip-the-academics-go-to-the-sources operation and propose some theories of my own that nobody else seem to have conjured up. If my theory is good, somebody really has, but I haven't discovered it yet. If my theory is bad, then nobody will ever accept my theory for being crack-pot balderdash.
Nevertheless: the source I'm boldly invoking is Papias of Hierapolis, speaking through the citation of Eusebius the Christian historian:
[John] The Elder used to say: Mark, in his capacity as Peter’s interpreter, wrote down accurately as many things as he recalled from memory—though not in an ordered form—of the things either said or done by the Lord. For he neither heard the Lord nor accompanied him, but later, as I said, Peter, who used to give his teachings in the form of chreiai, but had no intention of providing an ordered arrangement of the logia of the Lord. Consequently Mark did nothing wrong when he wrote down some individual items just as he related them from memory. For he made it his one concern not to omit anything he had heard or to falsify anything.
Naïve interpretation (according to wikipedia on chreia): John Mark wrote a list of logia like the Gospel of Thomas, of the form followers asks, Jesus answers, or Jesus says, without context in either case. Mark wrote the logia in the order that he remembered them, and he wrote them down remembering Simon Kaifas (Peter) speeches. If this text is to be taken for true, then John Mark didn't write the Gospel of Mark, at the very best he wrote fragments that later was extended and reordered to become the "Gospel of Mark".
Furthermore Papias writes:
Therefore Matthew put the logia in an ordered arrangement in the Hebrew language, but each person interpreted them as best he could.
The only clear thing here is that Matthew also wrote something like the Gospel of Thomas, but his logia were 1. written in some order, possibly chronological, or perhaps alphabethical, 2. wrote them in the Hebrew language. Scholars use to assume that Matthew wrote in Greek, but I think this is unwarranted, given that Eusebius's citation of Papias is correct, and that Papias correctly relates to truthful sources that know what they speak about.
My bold addition to this scheme is that Papias himself either collated the "modern" Gospel of Mark or the "modern" Gospel of Matthew, or both. He wrote a (now lost) five book comment on the Gospels according to later sources. He also interviewed a lot of elders, including the daughters of Philip. It is reasonable to believe that he collated the original Gospels of Matthew and Mark with the stories of his interviewed persons, so as to bring the original Gospels in chronological order and make reasonable interpretations of the chreiai and the logia. This constituted what was initially called the "Memoirs of the Apostles", but which later came to develop to the synoptic Gospels.