Why weren't the writings of the apostolic fathers (Clement I, Polycarp, Ignatius) included in the New Testament?

In short, because they wrote later, they were not apostles, their writings are clearly of a different literary quality and style to the new testament documents, and they were never widely treated as authoritative in the way the NT docs were. Simply put they were, mostly, never 'contenders' for inclusion

edit:

Sorry I haven't responded to some comments, I've been asleep. I've expanded my answer to cover some of these issues in more depth:

By the time of the apostolic fathers you see those texts referencing or in some cases quoting NT texts as authoritative. You see this, for example, in Polycarp’s letter to the Philippians, in 2 Clement (2.4), in the Letter of Barnabas (4.14). Justin Martyr refers to the ‘memoirs’ of the apostles and records that they were read in Church along with the readings of the Prophets (i.e. OT writings).
Once you get to a stage where the church is trying to work out the limits of the Canon, you get the following kind of criteria in play:

a. did they consider it inspired?

b. was it apostolic? i.e. does it purport and is believed to be written by an apostle or someone closely connected with an apostle (i.e. Mark’s traditional association with peter). The Muratorian Canon fragment notes that Shepherd of Hermas is non Apostolic and so not to be accepted; at the same time he recognises that Luke had not encountered Jesus as a living eye-witness.

c. was it old enough? simply put the texts needed to derive from that apostolic period, not a subsequent 2nd to 3rd generation period.

d. the text should generally be considered applicable to the whole church, not a private correspondence.

e. the texts in view were “for reading in the church”; vice versa, canonisation was a process of recognising what texts were already being used in churches.

f. doctrinal considerations: the teaching of the text needed to cohere with the church’s view of orthodoxy. Obviously the challenge of Marcion and his followers makes this crucial for churches. While there is some circularity here – how do you establish correct teaching if not from texts, how do you judge correct texts, if not from teaching that derives from texts, it gives you a sense that just because a text was accepted elsewhere and had an apostle’s name attached, did not mean naïve acceptance. documents were weighed against the already clear Rule of Faith, the core of the church’s message as handed down and as seen in the already accepted canonical texts.

From 170AD onwards you get actual ‘lists’ appearing, such as the Muratorian Canon. It’s a little bit fragmentary. What’s interesting about it is (a) it inclused Wisdom of Solomon and Revelation of Peter as canonical, (b) it doesn’t include Hebrews, James, 1 & 2 Peter, and 3 John, and (c) it clearly rejects some other texts like Paul’s Letter to the Laodiceans.
It’s interesting to look at Eusebius’ list from his Ecclesiastical History 3.3. Eusebius lists books as accepted, disputed, spurious, and heretical. But he lists very few apostolic fathers. The Shepherd of Hermas, Didache, and Letter of Barnabas are all there, but in the Spurious category. These are the kinds of texts that might make a claim to being legitimate, but were not considered so. The other types of spurios and heretical texts tend to be other Gospels and Acts. They are not the other elements of the AF collection.
367 AD is the first time you get a document that lists the NT canon in a form recognisably the same as that accepted widely today of 27 books. It’s very likely that many church communities used something close to this as their functional NT canon earlier, but 367 is just the earliest document that attests to it. It’s Athanasius’ 39th Letter, on the occasion of Easter. This canon was ratified at the Council of Hippo in 393, the 3rd Council of Carthage 397, and the 4th Council of Carthage in 419.