They were either written by an apostle, or under their supervision and approval.
Either way, those books were quickly recognized as canonical.
Wrong. When it comes to Revelation and Jude there is valid speculation that the John of Patmos differs from John the Apostle and that Jude's author was not the apostle.
Wrong again. Only the 4 Gospels were universally recognized at the start
Did you know that for most of the Early Christians until the canon was closed in the third century, for most of these, the book of Revelation in fact was NOT in the Bible?
Christian canons start from Marcion's in the first century which tossed out the entire OT and kept only the Pauline Epistles and the Gospel of Luke.
The New Testament was not complete and clear-cut during Apostolic times, indeed, this was not finalized until the 4th century.
in the New Testament itself there is some evidence of a certain diffusion of canonical books: II Peter, iii, 15, 16, supposes its readers to be acquainted with some of St. Paul's Epistles; St. John's Gospel implicitly presupposes the existence of the Synoptics (Matthew, Mark, and Luke). There are no indications in the New Testament of a systematic plan for the distribution of the Apostolic compositions, any more than there is of a definite new Canon bequeathed by the Apostles to the Church, or of a strong self-witness to Divine inspiration.
I repeat -- Only the 4 Gospels were universally recognized at the start -- Irenæus, in his work "Against Heresies" (A.D. 182-88), testifies to the existence of a Tetramorph, or Quadriform Gospel: Matthew, Mark, Luke and John. This is attested to by the Didache (70 AD) and the Epistle of Barnabus
Now the Pauline Epistles -- the majority were also universally recognized as canon, with the exception of Hebrews which was not universally recognized, indeed left out of many canons like the Muratorian Canon.
In fact take the example of the Eastern Syriac Church which left out the Apocalypse and James, Jude, I and II Peter, and the three of John
So your statement is incorrect
Quite wrong, the level of historical revisionism among some is incredible.
Origen in the 200s travelled the Christian world and indicated that the ONLY books of the New Testament of the Bible (note again the Bible is a collection of Books) were the Gospels, the thirteen Pauline Epistles, Acts, Apocalypse, I Peter, and I John
The contested writings were Hebrews, II Peter, II and III John, James, Jude,