Literary independence of the synoptics is certainly a minority viewpoint (and has been for centuries—Augustine argued for dependence in “Harmony of the Gospels”), but it exists.
I am unaware of any ancient scholars arguing for literary independence. A few modern scholars are cited here.
Perhaps the most noteworthy example in recent years is that of Eta Linnemann, who published “Is There a Synoptic Problem” in an effort to reconsider commonly held views. Linnemann comes out against the popular Two-Source hypothesis and explores the possibility that none of the Gospel authors directly copied from another. Her critique of pre-suppositions in Gospel scholarship in general and hyper-skepticism specifically have garnered positive reviews from those who believe scholarship is/was off-course, and have been frustrating to those who wish to hold to the conclusions of early 20th century scholarship.
John AT Robinson expressed a similar sentiment in “Redating the New Testament” (see p. 314), concerned that questions are sometimes settled too quickly and the answers are not re-examined:
“[O]ne cannot help observing how the caravan of New Testament
scholarship has tended to move on to fresh sites...leaving behind the
other workings in the state in which they then were and taking over
their results to date, without further reexamination, as the
presuppositions of the new work. This I suggested happened, notably in
regard to the synoptic problem, both when the form critics took over
from the source critics and when the redaction critics took over from
the form critics. These newer disciplines have had much illumination
to shed. But they become dangerously unrelated to reality when they
ignore detailed investigation of the 'introductory' questions - and
still more when they claim that these can be settled by the answers to
their own questions.”
In this sense the theory may be marginalized not because it is unsupported by evidence, but because it isn’t taught in the textbooks.
Rainer Riesner argues for a somewhat complicated relationship among the synoptics, in which they rely on common oral and written sources, but none directly rely on each other. Riesner draws attention to the powerful role of memory in ancient times, and suggests that while there were written sources before the synoptics, much of the common material is the result of memorization and careful tradition-bearers. (a summary of his views are found in chapter 5 here)
Riesner actually uses Luke’s preface to argue against literary dependence:
“Some scholars emphasize anew the role of eyewitnesses and
earwitnesses as being the tradition bearers and informants of the
Gospel writers [such as Bauckham or Hengel]. Indeed, in the manner of
a historian of antiquity Luke in the preface to his Gospel claims that
he received the traditions of such primary witnesses. And in Romans
10:14-17 Paul gives an outline of a chain of tradition reaching from
‘the word of Christ’ through the apostles to their hearers.” (“The
Synoptic Problem – Four Views” [hereafter “Four Views”] - p. 102)
And Riesner points out that if Jesus’ sayings were considered on par with Old Testament teachings, we should very much expect that they were memorized and handed down with accuracy:
“The shorter or longer poetically structured sayings of the Old
Testament were already memorized…[t]o his sympathizers who believed in
Jesus’s prophetic or even messianic authority, his sayings were of the
utmost importance.” (Four Views pp. 102-103)
Riesner and others have noted that an itinerant preacher like Jesus almost certainly gave the same sermon more than once, affording many opportunities for hearers to learn His words through repetition. While Riesner does not rule out common written sources for the synoptics, he certainly draws attention to the fact that modern cultures are quick to downplay the role and power of memorization in antiquity. His assessment of the evidence is:
“When one understands the role of oral tradition and eyewitness memory
in the first century…a different picture emerges. Rather than being
dependent on one another, the three Synoptic Gospel writers each
partially used the same intermediary sources, both oral and written.”
(Four Views p. 90)
Why does this remain a minority viewpoint?
Tremendous correspondence in Greek
The close correspondence of the synoptic gospels in Greek is much more impressive than is readily apparent in English. Greek words have suffixes applied to indicate their relationship with other words and as such, word order is not so crucial to meaning in Greek as it is in English. When a Greek text has identical (or nearly identical) order of material, grammar, and word order, it suggests a literary relationship.
David Barrett Peabody highlighted Matthew 3:7-10 & Luke 3:7-9, during a portion of which “there is verbatim agreement between Matthew’s and Luke’s Gospels for a total of sixty-two words in Greek, in the same word order and in the same Greek grammatical forms” (Four Views p. 68). Hundreds of other (shorter) examples could be cited.
The minor agreements of Matthew & Luke against Mark (see here) are not only problematic for the Two-Source Hypothesis, they present some difficulty for literary independence. That Matthew & Luke would share precisely the same small details hundreds of times is readily explained if one used the other as a source; it appears terribly improbable otherwise.
Luke’s prologue and the Argument from Order (my thoughts on the latter topic here) could each support their own lengthy discussion in favor of a literary relationship.
While I applaud the scholarly effort to reopen questions that have been prematurely “closed” (same thing happens in scholarly literature as sometimes happens on stackexchange =) ), ultimately, I do not find the literary independence hypothesis convincing.
My own study of the synoptic problem (see here & here) has led me to the conclusion that oral transmission certainly played a role, but the similarities not just in stories, but in order, wording, and Greek grammar, are best explained by a literary relationship.