2 Clem is late and not an important source - it's rarely mentioned or translated into English:
As far as I've been able to tell it isn't quoted in the early to mid 2nd Cent time period when early writers were quoting other gospels (though there's a possible exception in 2 Clement -- but its date isn't well established either, and the quotation could have come from another source).
The Gospel of Thomas isn't mentioned even once in one of the latest books I've acquired: The making of Paul - Constructions of the Apostle in Early Christianity (2010). Richard I. Pervo. Fortress Press.
I would date it about 120-125 C.E., too late to be included in the core of the New Testament anywhere.
My theory is that Christians/"Gnostics" really needed a Bible by 95 C.E., so they included what they had at hand by that time, books that had been circulated for some time - and since papyrus scrolls was destroyed very quickly when in use and a whole lot of generations of copies already had been made, they thought that all these were old:
Mark
Romans
1 Corinthians
2 Corinthians
Galatians
Ephesians
Philippians
Colossians
1 Thessalonians
Philemon
Hebrews
James
1 Peter
1 John
Jude
2 Thess and especially it's views (hyper-Paul) had been in use for a few decades but when 2 Thess was written ~80 C.E. the letter was anti-Paul, and therefore not accepted by this generation who favoured Paul.
I think other large Gospels like Mt and Lk didn't find their final form until ~9-11 years later - i.e. they were known but not canonized and were being edited, by whom is a separate question. Lk was canonized thanks to Marcion of Sinope and used unabridged after he died. Even so Lk is not too far from the time of Jesus because it had started to be written right after the Epistles of Eph and 1 Pt and had two ancient sources.
Jn was known but not yet in widespread use until 97 C.E. and onwards, that is my theory. My theory is that Jn became used much more because of revived interest in written accounts thanks to Christianity definitely becoming a documented religion a few years earlier.
It's no wonder that there was a need for more Gospels than Mk and that the other Gospels don't add too much.
Personally, I add only 4 Ezra and 1 Clem to the above canon of the New Testament. I read Lk, except chapters 1-2 and Jn except verses
7:53-8:11 (follow link to Accordance forums!) and chapter 21, as canonical sources. I don't read 2 Cor 10-13, Colossians and Hebrews 9:15-13:25.