The primary argument against the traditional assignation of authorship of the Gospels is that they're not referred to by name in the Church Fathers before Irenaus (~180 AD). But by the literary standards of the ancient world, that just doesn't seem that weird to me? The early Fathers were referring to works by authors in living memory, and the prevailing bibliographical standards were somewhat lax for even further removed sources. We wouldn't claim that anyone who referred to Aristotle as "the philosopher" or Isaiah as "the prophet", as often happened, didn't know their names. Moreover, Justin Martyr (~150 CE) is often cited as evidence against named Gospels, but in the same source he refers to the Gospels collectively as "the memoirs of the Apostles", which doesn't help the case for anonymous authorship.
The biggest problem for the anonymous theory, however, is simply that there are no manuscript witnesses to support it: all the early manuscripts have titles. So the anonymous theory has to posit:
- The last Gospel, John, was reduced to writing about ~95 AD.
- The Gospels circulated throughout the Mediterranean in anonymous form.
- At some point before 180 AD, the Church decided to get its story straight and assigned names to the Gospels.
- At that point, everyone from France to India started referring to the Gospels with their present names, without any controversies or mixups.
- All of the earlier anonymous manuscripts were lost to history.
I'm not saying that definitely could not have happened, just that it doesn't seem especially more plausible than the traditional account.
I've read a suspiciously large amount of historical criticism for someone who doesn't do this for a living, thinking I would finally get to the bottom of what the New Testament was "really" about. I came away with the impression that the optimal amount of attention to pay to NT hist crit is either a lot, or zero. Every generation of hist crit somehow comes away with the conclusion that the NT is really about the issues of concern to that generation. In fact this has been going on ever since the field was founded by 19th century German Romantics, who discovered that the NT was the product of national ur-spirits expressed through folklore [!]
Our cultural familiarity with the NT sometimes keeps us from seeing how strange a collection of documents it really is. It is perhaps the best attested collection of sources in the ancient world, yet contains a mixture of Greco-Roman biography and history, supernatural events, and mystical theology. I am not trying to persuade you of any particular view about NT scholarship so much as challenge the idea that there's anything cut and dry about it.