The Mishne and Gemara discuss categories of halacha for which there is very little scriptural basis from the Torah. Shouldn't shechita and the laws of kosher and treife have been included in the list? It seems to me that the whole of shechita and all of the voluminous details of kosher and treife is based on Torah She'baal Peh (oral law) and there is only one posuk (verse) that alludes to it.
There appears in fact to be four categories:
a) Porchin b'Avir v'Ein Lahem Al Mah sh'Yismechu -- laws that "hover in the air" with no scriptural base without the Oral Law's help.
b) Harerin ha'Teluyin be'Se'arah -- a clear Scriptural precept (Shabbos, Chagigah, Me'ilah) clearly mentioned in the Torah, but their detailed Halachos are "hanging by a hair, supported only by slight allusions in the verse, and there are many Halachos."
c) Yesh Lahen Al Mah sh'Yismechu -- laws which are explicit in the Torah, but there is one instance where it is not clear without the interpretation of the Oral Law.
d) The fourth category, which is not mentioned in the Mishnah here but which is that which does not conform with the above three categories, is where there is a clear Scriptural basis but no clarification therein and which is entirely elaborated by the Oral Law. This would include the majority of Mitzvos, such as Shechitah, Kashrus, Tefilin (and Tzitzis, perhaps, unless this is already included in the third category), Bris Milah, etc. In these cases, we know that the basic Mitzvah exists from the Torah itself (unlike the first category). We do not know the details from the Torah (unlike the third category), and they are not "hanging by a hair" (unlike the second category). They are known to us by being passed down from generation to generation and from Halachah l'Moshe mi'Sinai, without much resort to Torah exegesis and hence they are outside the ambit of the Mishnah.
Perhaps the Mishnah mentions only Shabbos, Chagigah, and the others to teach a Chidush -- that even these areas of Halachah have many significant Halachos that are learned only from a slight allusion. Shechitah, though, is obvious that it has very little details supported by verses, because all of its Halachos were not written in the Torah, but were passed to Moshe Rabeinu and transmitted orally.
I do not think that your answer differs materially from my answer regarding the fourth category of Halachos. Is not this the self-same point?
Yes, you are correct. We forgot to delete our answer from the copy we sent to you, after receiving your comprehensive answer!