The gemorah discusses a case when the Kohen stays outside the Azorah without Tvilah and uses an elongated knife for Shechitah.....
Let me quote the Gemoroh:
"Tiboyah Lerabonan........Aval hachah dekoavid avodah..." - ".....but here (with a long knife) where he (the Kohen) is doing an avodah (of Shechitah) the Rabonan would not allow it without Tvilah first...!"
From here we see that Shechitah of the Korban is called an "Avodah".....(even if permitted by a non-kohen).
Now turn back on Daf to 30A and look at the second toysfos down the page (Divrey Mas'chil: "Ain Odom Nechnas...") where toysfos says that ".....Afiloh Shechitah Delav Avoidah hee..." that Shechitah is Not called an Avoidah ???
Now is Shechitah an Avoidah or not?
Maybe Toysfos mean that Shechitah is not an Avoidah where a Kohen is reqired...? (I know that Shechitah is 1 of the 4 (3 are needed, 4th is "halichah, but one can do without) Avoidos needed for each Korban, but Shechitah is Kosher for a Zor, a Non-Kohen.
You were "Mekavein" to the Rishonim with your question and answer. They elaborate on your answer in different ways:
1. TOSFOS YESHANIM 31a DH AVAL writes that Shechitah is not an Avodah so that it should require a Kohen like the other Avodos on the blood do (as you answered). Note 2 at the bottom of the page adds that it is, however, considered an Avodah in as much as it requires a lottery (as MISHNAH above 25a states that this was part of the second "Payis")
(See Gemara above, 27b, that the reason Shechitah requires a Payis, even though a non-Kohen is not culpable for the death penalty for performing it, is because Shechitah is the start of the Avodah. See also RITVA there)
2. TOSFOS ROSH (end, DH MAHU) explains that even though Shechitah does not require a Kohen, nevertheless, relatively speaking, it is considered more of an Avodah than the sprinkling of the blood on the thumbs of the Metzorah. Therefore, even though according to Rabanan, Tevilah is not necessary before sprinkling the blood, nevertheless Tevilah is necessary before Shechitah.