Here I'll be looking at Dr Carrier's article "How We Can Know 1 Clement Was Actually Written in the 60s AD", and comments in the article about 1 Clement's use of quotes from Jesus. While Carrier's article is about the dating of 1 Clement, the question of the date that 1 Clement was written isn't a concern here.
Carrier's article is here: https://www.richardcarrier.info/archives/22313
Some of the comments Carrier makes about Gospels (note that Carrier also quotes himself from his book "On the Historicity of Jesus":
Note that Carrier means the written Gospels when he argues that 1 Clement has no knowledge of the Gospels. From this, he concludes:Finally, 1 Clement knows nothing in the Gospels—evidently, those stories hadn’t been invented yet...
1 Clement “never once places Jesus in history or ever tells any stories about him, never uses his stories as an example for anything (despite the letter being a long series of arguments by example), nor ever quotes anything Jesus says in the Gospels” even when it would clinch several of its arguments (OHJ, p. 309). Indeed, “of all the dozens of stories Clement summarizes as examples for Christians to follow, all come from” the Old Testament and recent martyrology, “none from any Gospel or anything in the life of Jesus” (Ibid., p. 314)...
I list many specific examples of Clement’s ignorance of the Gospels in OHJ (Ch. 8.5): [Note: I've snipped some of them - GDon]
* Sayings of Jesus are quoted, but never anything from the Gospels; rather they simply quote the Old Testament (see The Original Scriptural Concept of ‘The Lord’ Jesus), as in §8, §22, §24-26, §30, §45, or otherwise unknown sayings.
* For example, Clement once quotes Jesus’s commandments not matching any Gospel (§13)—and even though each commandment on his list is expanded into more elaborate teachings, parables and stories in the Gospels, Clement never uses or references any of them.
* Clement also quotes the line from the Psalms as about Jesus, “Thou art my Son, today have I begotten Thee” (§36, §59), yet he has no evident knowledge this comes from a story of his Baptism; he only knows of it from the Psalms.
* Even the lone exception evinces no knowledge of the Gospels: the “Woe to that man!” saying Clement has no idea of ever being connected to Judas (§46), but instead knows only as a preface to the “millstone” prophecy, which the Gospels relocate to an entirely different place and story, which Clement also exhibits no knowledge of (OHJ, pp. 311–12).
Here are passages from 1 Clement, followed by similar passages found within the Gospels:1 Clement’s complete ignorance of the contents of the Gospels and the Jewish War and its outcomes unquestionably dates it prior to both; while its mention of Paul’s recent death ensures it post-dates his authentic letters, which were completed by the end of the 50s, leaving the early 60s as the only possible date of 1 Clement’s writing.
... being especially mindful of the words of the Lord Jesus which He spoke, teaching us meekness and long-suffering. For thus He spoke: "Be merciful, that you may obtain mercy; forgive, that it may be forgiven to you; as you do, so shall it be done to you; as you judge, so shall you be judged; as you are kind, so shall kindness be shown to you; with what measure you measure, with the same it shall be measured to you." By this precept and by these rules let us establish ourselves, that we walk with all humility in obedience to His holy words. For the holy word says, "On whom shall I look, but on him that is meek and peaceable, and who trembles at My words?"
Luke 6:36 Be merciful, just as your Father is merciful. 37 Do not judge, and you will not be judged. Do not condemn, and you will not be condemned. Forgive, and you will be forgiven. 38Give, and it will be given to you. A good measure, pressed down, shaken together, and running over will be poured into your lap. For with the measure you use, it will be measured back to you.”…
Matt 7:1 “Do not judge, or you will be judged. 2For with the same judgment you pronounce, you will be judged; and with the measure you use, it will be measured to you.…
Mar 4:24 And he said unto them, Take heed what ye hear: with what measure ye mete, it shall be measured to you: and unto you that hear shall more be given.
Remember the words of our Lord Jesus Christ, how He said, "Woe to that man [by whom offences come]! It were better for him that he had never been born, than that he should cast a stumbling-block before one of my elect. Yea, it were better for him that a millstone should be hung about [his neck], and he should be sunk in the depths of the sea, than that he should cast a stumbling-block before one of my little ones.
Mark 14:21 but woe to that man by whom the Son of man is betrayed! good were it for that man if he had never been born.
Mark 9:42 And whosoever shall offend one of these little ones that believe in me, it is better for him that a millstone were hanged about his neck, and he were cast into the sea.
Our apostles also knew, through our Lord Jesus Christ, and there would be strife on account of the office of the episcopate. For this reason, therefore, inasmuch as they had obtained a perfect fore-knowledge of this, they appointed those [ministers] already mentioned
Mark 13:22 For false Christs and false prophets will appear and perform signs and wonders that would deceive even the elect, if that were possible.
23 So be on your guard; I have told you everything in advance.
Jacob, through reason of his brother, went forth with humility from his own land, and came to Laban and served him; and there was given to him the sceptre of the twelve tribes of Israel... from him have sprung the priests and all the Levites who minister at the altar of God. From him also [was descended] our Lord Jesus Christ according to the flesh.
Mat 1:2 Abraham begat Isaac; and Isaac begat Jacob; and Jacob begat Judas and his brethren;
Luk 3:34 Which was the son of Jacob, which was the son of Isaac, which was the son of Abraham, which was the son of Thara, which was the son of Nachor
Your mileage may vary with regards to the strengths of the parallels. Note that Carrier himself attempts to explain why those quotes aren't coming from written Gospels. But if they aren't quotes from the written Gospels, then they appear to be coming from a common source. 1 Clement mentions "Gospel" twice:
The apostles have preached the Gospel to us from the Lord Jesus Christ; Jesus Christ [has done so] from God. Christ therefore was sent forth by God, and the apostles by Christ. Both these appointments, then, were made in an orderly way, according to the will of God...
Take up the epistle of the blessed Apostle Paul. What did he write to you at the time when the Gospel first began to be preached? Truly, under the inspiration of the Spirit, he wrote to you concerning himself, and Cephas, and Apollos, because even then parties had been formed among you.
Take up the epistle of the blessed Apostle Paul. What did he write to you at the time when the Gospel first began to be preached? Truly, under the inspiration of the Spirit, he wrote to you concerning himself, and Cephas, and Apollos, because even then parties had been formed among you.
"Gospel" here appears to mean the message rather than written texts. So it may mean oral traditions, or written texts that are yet to be called "Gospels". If, as Carrier believes, the Gospels hadn't been written yet, it suggests to me good evidence for oral tradition.
I'll update my notes. I got a bit confused with the quote in 1 Clement. The full quote is: