“scattered all over the place”
is the definition of NO CANON!!!
In regards to the NT, what was finally settled upon (after sorting out later arising writings) but that which the Apostles as direct witnesses wrote, along with the writer of Luke and Acts who was amid and living among those first witnesses, if not one himself also?
In that way the NT was more imposed upon the early church as soon as we look beyond those original disciples in defining who or what is the church.
The NT texts were well known among all, from the earliest beginnings-- even soon after the various epistles of Paul were written, for those did circulate among the many.
As to that which there was ever much doubt (looking at the contents of NT as we know it today) is John's Revelation (not fully accepted by all, from it's first circulating among the various congregation) and the epistle to the Hebrews, with this last having not been as well circulated among the further-from Israel locales, in comparison to the Gospels, Paul's other letters, those few of Peter, James & Jude, etc. Which reminds me...some were hesitant towards the small book of Jude, yet that fairly early became to be included also.
Here we have the core -- from earliest times.
It was not those of Rome who "defined" the canon much at all, for if one is to investigate the formation of canon, other than the very early Tertullian--- writers of the "Latin" church, in regards to issues of canon were rather on the sidelines until centuries later.
And again--- what became inclusive of NT canon but that which was indeed imposed upon the church?
It does no good to now to point to Rome as some source for that, for they (those of the Latin branch of the church) were themselves recipients as were all others --- other than the original Apostles, and perhaps Luke if that man was not one who had witnessed Christ personally & directly himself.