How much did Irenaeus influence our view of the gospels?

Critics often suggest that Irenaeus had an inordinately large influence on what gospels were considered canonical, what authors those gospels were attributed to, and other gospel issues. However:

"Irenaeus hardly adopted precisely these four Gospels randomly, especially given his emphasis on church tradition; and is it an accident that he chose the four Gospels more reflective of first-century Judean traditions than our other extant gospels (the 'apocryphal' gospels and gnostic sayings treatises)?" (Craig Keener, Acts: An Exegetical Commentary, Volume I [Grand Rapids, Michigan: Baker Academic, 2012], 399)

Martin Hengel mentioned a line of evidence that's rarely discussed:

"Claus Thornton has shown that this [a passage in Irenaeus about gospel authorship] is an earlier tradition, which must be taken seriously; as the geographical references and references to persons show, it is written throughout from a Roman perspective....As Thornton has demonstrated, it corresponds to the short notes about authors in the catalogues of ancient libraries, of the kind that we know, say, from the Museion in Alexandria. Presumably this information comes from the Roman church archive." (The Four Gospels And The One Gospel Of Jesus Christ [Harrisburg, Pennsylvania: Trinity Press International, 2000], 35-36)

Here's the passage in question:

"Matthew also issued a written Gospel among the Hebrews in their own dialect, while Peter and Paul were preaching at Rome, and laying the foundations of the Church. After their departure, Mark, the disciple and interpreter of Peter, did also hand down to us in writing what had been preached by Peter. Luke also, the companion of Paul, recorded in a book the Gospel preached by him. Afterwards, John, the disciple of the Lord, who also had leaned upon His breast, did himself publish a Gospel during his residence at Ephesus in Asia." (Against Heresies, 3:1:1)

Notice how unnecessary the reference to Peter and Paul's work in Rome is. I don't recall anybody else describing the timing of the composition of the gospel of Matthew that way. Similarly, connecting the origins of Mark's gospel to the apostles' work in Rome ("After their departure") is unnecessary. Just before what I've quoted above, Irenaeus refers to how the apostles had spread the gospel "to the ends of the earth", so the shift to such a focus on Rome is somewhat contrary to the context. Irenaeus probably was citing a Roman source along the lines of what Hengel refers to above. So, Irenaeus is citing an earlier source that presumably made its claims independently of Irenaeus, a source that was well positioned to have significantly reliable information (the Roman church).

For more about how Irenaeus' influence is often overestimated in these contexts, see here, here, and here.