The tofeta speaks of 'christian writings'. There is no way they would have been burning the Hebrew scriptures. The christian writings were the 4 gospels and the letters of the apostles. These were all complete and in circulation before the end of the first century.
Yet we Christians had books in our Old Testament that the Jews later came to reject, because they proved Christ's Messiahship. Plus, different churches had different lists of what they considered to be in the New Testament--for example, Antioch had 22 books in their New Testament, while Alexandria had 33. It would also take centuries before the Gospels (known to the first Christians as simply the "memoirs of the Apostles") would be considered as Scripture, and longer still before they were regarded on the same level as the Old Testament. The New Testament as we know it today did not exist in the minds of the first Christians.
Also, the Jews have been known to burn even the writings of the prophets when they don't like what they hear--remember when King Jehoiakim of Judah burned the scroll of Jeremiah in Jeremiah 36? The Jews would have had no problem burning works which supported the claims of who they thought was a false Messiah, and shutting these same works out of the final cut of their canon. This is why the Jews don't have the Books of the Maccabees, the Wisdom of Sirach, the Wisdom of Solomon, the Prayer of Manasseh, various parts of the Book of Daniel (such as the Song of the Three Youths, the story of Susanna, the story of Bel and the Dragon,) the Epistle of Jeremiah, or the books of Tobit, Judith or Baruch. This is why they began standardizing the Tanakh and getting rid of all the textual variants, because we were doing too good with proving Christ's Messiahship through the Septuagint and the Hebrew variations which gave rise to it.
If they weren't, how were they being collected up and burned? Obviously they were already well established.
They were circulating among local groups--you wouldn't find the epistle to the Corinthians in Smyrna, for example--but I wouldn't call them "established." Not even the Four Gospels were each spread out across the entire Christian world until much later.
The Name of God appears in its abbreviated form in the greek scriptures. Why would they use the abbreviated form at all if they did not know what it was?
They knew the abbreviated form. How to pronounce that abbreviated form is what was forgotten. The vowel notations in the Hebrew text that we have today were medieval inventions by the Masoretes to ensure that there was only one way the Hebrew Scriptures could be read--the least Christian way possible.
does not using it help people come to know who God is?
I don't think calling Him "Jehovah" necessarily helps a whole lot more--yes, we can explain the etymology of it as being "I Am Who I Am" or "I shall be Who I shall be", and further clarify this as meaning that God is causeless and wholly unbound, and that He alone is fully self-sufficient and complete. But we can explain this without using the medieval rendering "Jehovah". YHWH works just as well--or heck, just cut straight to the translation. The main point of Who God is isn't that His name is YHWH. It's that He is the Creator of all, the Lord and Judge of all, Lover of mankind, our Physician and our Fortress, our Rock and our Refuge, our Liberator and our Life, our Healer and our Helper, our Savior and our Sanctifier, our Redeemer and our Ruler. He is Holy Trinity, the Father, the Son and the Holy Spirit, One in essence and undivided. And I don't need to tell you that His name is YHWH to get any of that across.