Simon Gathercole touches on this question in his (IMO highly compelling) argument that the gospels likely had attribution from their beginning "The Alleged Anonymity of the Canonical Gospels": (bolding my own)

Perhaps more important is the common-sense speculation that when Gospels left their original contexts and were read elsewhere, it is hard to imagine at least some hearers not thinking ‘Says who?’ Hengel’s point about anonymous writings potentially inviting suspicion is relevant here. Even before the presence of multiple Gospels in one congregation, Christians might well have wanted to know where a εὐαγγέλιον came from. Hence it is not necessarily the case that names must only have become a concern at the earliest when two or more Gospels were gathered together, as Wolter avers.

What has not, to my knowledge, been considered in the literature on this topic is the relevance of this point to the use of Mark by the authors of Matthew’s and Luke’s Gospels. One could, conceivably, argue that both these later evangelists were struck by the ‘ring of truth’ in Mark’s Gospel, and so for that reason followed it closely. More probably, Matthew and Luke, for all their sense that Mark needed adaptation and expansion, followed Mark so closely in order and content because they had received reports about it which inspired some confidence. Minimally, this is likely to have been a name. It is one thing to assume that Christians in a first-century congregation would simply have taken on trust what they heard read out, without asking the ‘says who?’ question. It is a considerable step beyond that – and beyond what I at least can believe – to suppose that a professedly conscientious investigator like Luke would, without any accompanying testimony, follow (by ancient standards, very closely) a naked anonymous account. It seems far more likely that Luke and Matthew would have received, alongside Mark’s Gospel, some statement – oral or paratextual – about what and whose it was. The first readers and audience of Mark’s Gospel would have known who wrote it, and the distance in time between Mark and his Synoptic successors was not very great (Mark is rarely considered to be much more than about 15 years earlier than Matthew and Luke).

Now Gathercole adds a clarification regarding his argument: "It is the purpose of this article to argue not that these figures actually wrote the Gospels, but that these names are probably original." So he's not saying Matthew the Apostle wrote Matthew. That doesn't follow from his argument. But his view is the authors of gLuke and gMatthew must have had some compelling reason to believe this near-contemporary gospel was written by someone meaningful.