I remain convinced that, somewhere along the line, the text has been badly redacted so that getting the proper meaning from the words is not longer really possible. We can, of course, make guesses but, of course, there is no way we can know if our guesses are right or not.
Why would you come to that conclusion?
The verse seems perfectly at home in Matthew. I shall explain. First I will show you this site: http://therefinersfire.org/
They are Jewish Christians, arguing that Torah law is still in power. They scorn Christians who have mangled Paul's words to suit the idea that Torah law has been replaced. I don't know how they get past Galatians (and I have sent them an email), but I will ignore that problem. It's theirs.
As you point out, Matthew didn't know about any other book in the NT, and thought he was writing to Jewish Christians; ie, the people like who I just linked to, that Paul scorns in Galatians. Would there have been plenty of time for redaction? (and correction). Not if Matthew was redacted a bit later, to suit Jewish sensibilities, or if the Jewish Christian church persisted so long that nobody could change Matthew, because it was now sacrosanct, and in multiple copies.
Without Matthew, the Christian doctrine would be pretty lame. I think if you cut it out, you are left with Luke, who is not even an eyewitness, because he says so. John is a different Messianic model. So, Christians had to include it, because it had authority, up to the time of Justin Martyr. The early church would have been full of Judaisers, whether the Pauline Christians liked it or not. So, their solution was to hide the problem.
The first "fulfill" is plerOsai -> complete
The second "fulfill" is genetai -> the universe has ended
The next "fulfill" is in John 19:28 tetelestai / teleiOthE -> finish/ perfected
Thus you have a load of words that are not etymologically connected, but are united by the translation, to create the preconception of what Pauline Christians want to see.
From the top; this is what I believe Matthew, the Jewish Christian says: (I will write my literal interpretation in purple)
 Think not that I am come to destroy the law, I haven't come to destroy the law, or even change existing law, because I'm a Jew, who believes his father. Who said that I did?
or the prophets: I am not come to destroy, but to fulfil.I am here to explain to you how to follow Jewish laws, because you have it all upside down
 For verily I say unto you, Till heaven and earth pass, one jot or one tittle shall in no wise pass from the law,The law is divine and shall exist for all time, and even in heaven.
till all be fulfilled.re-iterates (tautology) Until God's whole plan has unfolded (including resurrection, and people in heaven eventually dying, after the eon is over)
 Whosoever therefore shall break one
of these least commandments, and shall teach men so, he shall be called the least in the kingdom of heaven:If anyone teaches you to break many Jewish laws, they will probably not make it into heaven
but whosoever shall do and teach them, the same shall be called great in the kingdom of heaven.(Read between the lines) It's OK to break some Jewish laws, but you must never teach people that it's OK to do so, or that you plan to do so, yourself
 For I say unto you, That except your righteousness shall exceed
the righteousness of the scribes and Pharisees, ye shall in no case enter into the kingdom of heaven.In order to enter the Kingdom, you must at bare minimum adhere to all Jewish law, as the scribes do, ... but you have to do even more than them... explained next
 Ye have heard that it was said by them of old time, Thou shalt not kill; and whosoever shall kill shall be in danger of the judgment:
 But I say unto you, That whosoever is angry with his brother without a cause shall be in danger of the judgment: and whosoever shall say to his brother, Raca, shall be in danger of the council: but whosoever shall say, Thou fool, shall be in danger of hell fire.
 Therefore if thou bring thy gift to the altar, and there rememberest that thy brother hath ought against thee;
 Leave there thy gift before the altar, and go thy way; first be reconciled to thy brother, and then come and offer thy gift.
 Agree with thine adversary quickly, whiles thou art in the way with him; lest at any time the adversary deliver thee to the judge, and the judge deliver thee to the officer, and thou be cast into prison.
 Verily I say unto thee, Thou shalt by no means come out thence, till thou hast paid the uttermost farthing.
...Now, this section is called the Sermon, and many have wondered if it is hyperbole, and whether you are supposed to adhere to all of it's impossible demands, to be a Christian. Even though, paradoxically, Jesus is supposed to forgive you for everything. Instead, they are missing that you are supposed to adhere to all of Judaism, and the Sermon is, to some extent, pick and choose.
What Jesus is saying is that you must adhere to Judaism, to be at least as good as the scribes, but you also need to attend to as many of the Sermon precepts as you can. This gives you the turbo boost, to get into the bottom of heaven. A person who makes his best shot to follow all of the Sermon, is aiming for the greatest in heaven.