Absolutely. But when was it decided that the "Gospel of Thomas," for instance, didn't make the cut? When, where, and by whom?
Could the Church put Thomas in the Canon?
No.
Why or why not?
Because it was never used Liturgically: Lex orandi, lex credendi. Nor was it referenced as authoritative by the Ante-Nicene Fathers, or received as Scripture in Orthodox/Catholic Churches. That's the basis the Ecumenical Councils used, to determine the authenticity of Scriptures: liturgical, patristic, and ecclesial Tradition. The spurious texts were not in the Tradition.
If it does, would Thomas gain in authority, or would it always have had this authority?
No other books can be added to or subtracted from the Canon of Scripture. This is on the authority of the Church, "the Pillar and the Foundation of Truth." 1 Timothy 3:15.
P.S. to Kolokotronis: from an Orthodox point of view, what say ye?
By the surviving history, it was never given any serious consideration.
Because it was never used Liturgically: Lex orandi, lex credendi. Nor was it referenced as authoritative by the Ante-Nicene Fathers, or received as Scripture in Orthodox/Catholic Churches. That's the basis the Ecumenical Councils used, to determine the authenticity of Scriptures: liturgical, patristic, and ecclesial Tradition. The spurious texts were not in the Tradition.
So take the question back a step. When the Church used a book, was it recognizing something that would have been true of the book in any case, or was it giving the book something? Could the Church have used Thomas liturgically, and if it had would Thomas be Canon?
"P.S. to Kolokotronis: from an Orthodox point of view, what say ye?"
I say I agree. God inspired The Church and that inspiration is manifested in Holy Tradition, what The Church always and everywhere believed. That was the infallible yardstick by which The Church measured the "contending" writings for inclusion in the canon of scripture. But truth be told, the only real difficulties presented were Hebrews and Revelations. As to the rest, by the 370s Christians knew what was "in" and what wasn't. For example, for Christians of the 4th century, the idea that the Gnostic "Gospel of Thomas" would be included in the canon was a laughable as it is today. And the reason for that is that The Church knows and teaches us what Christians have always and everywhere believed in this regard.