With a beloved skype chavrusa I am learning Zevachim BS"D!!
Here are some Zevachim thoughts dedicated in his zchus [he is quiet and humble so I keep his name private - but Hashem knows]:
The Gemara [Chullin 12] wonders whether a kotton can shecht Kodshim. Rashi explains that the issue is that he can't have kavana לשמה - for the sake of the korban. Tosfos says that a kotton is considered a מתעסק - preoccupied, and unable to have proper kavana.
Fregt Rebbi Akiva Eiger [if you follow the blog consistently then this question will have a very familiar ring...] - How come the gemara in Bechoros [7th perek] says that a cheresh [deaf-mute who lacks דעת] and shoteh are not allowed to perform the avodah because they are אינו שוה בזרעו של אהרן - not common to all of Aharon's descendents [most Kohanim are not חרשים and שוטים] and as a result, they are not מחלל the avoda [עיי"ש]. Why don't we just say that they are disqualified ["paseled"] because like Tosfos said about a kotton - they are מתעסק! That would disqualify them not only from the avodah but from shechitah [which is not an avoda] and they would also be מחלל the avoda. [לפי רש"י לא קשה שהרי חסרון לשמה אינו פוסל].
Moreover - the Gemara in Chullin [24b] derives from a pasuk that a kotton is pasul for avoda. Asked the Minchas Baruch and the Gra"ch [the Rov - not the doctor] - Why do we need a pasuk? Teipuk ley [Spanish for "answer instead"] that everything they do is מתעסק like Tosfos said!!??
The Minchas Baruch and Rav Chaim Ozer answered that מתעסק is only a פסול in shechita so the Gemara had to provide a source that they are pasul for all avodos.
But that is not universal. Many differ and hold that מתעסק disqualifies all the avodos.
So back to square one.
The Brisker Rov explained that since we "pasel" מתעסק for shechita in addition to all the avodos, it must be that it is a din in the עבודת הקרבן and not בעבודת הכהן. So that means that it wouldn't pasel הטבת הנרות and תרומת הדשן. So we need alternative sources to pasel and cheresh, shoteh and kotton in הטבת הנרות and תרומת הדשן as well.
Farentfert the two kashyas [translation - ding-ding! Correct answers!]
The Chazon Ish answered Rebbi Akiva Eiger's that there is a היכי תימצי where we would need the Gemara's
source of אינו שוה בזרעו של אהרן and we couldn't say מתעסק. Namely - a shoteh who is עתים חלים ועתים שוטה, sometimes sane, sometimes insane. In his sane moments he would not be a מתעסק but would be pasul alts אינו שוה בזרעו של אהרן. Also - a חרש who is speaks but can't hear which would make him a person of דעת. For such a person we need the source of אינו שוה בזרעו של אהרן.
He answers noch a terutz [Portugese for "You gotta live in Bnei Brak to think up such things"] that there is a nafka minah regarding one who sprinkles the blood towards the mizbeach and before it reaches "home base" he becomes a shoteh [the Gemara on דף ט"ו talks about one who does the sprinkling and before it hits the mizbeach his arm is cut off עיי"ש]. As far as being a מתעסק is concerned - we have no concern because when he did the זריקה he was one hundred percent sane but we have a problem מצד the issue of אינו שוה בזרעו של אהרן.
Another possible nafka minah is offered by the Tiferes Yisrael [in his חומר בקודש]: When he perform the avodah with a kosher, valid person. If the issue is מתעסק then it doesn't matter because he has a partner who is not מתעסק so SOMEBODY involved is halachically considered attentive and everything is fine. But if the issue is אינו שוה בזרעו של אהרן or [in the case of a kotton] a special pasuk that pasels - it doesn't matter that someone else is with him and the avodah is pasul.
HaGaon Rav Avraham Noach Garboz Shlita [מנחת אברהם עמ' מ"ג] points out that this terutz is untenable. If a pasul person and a kosher person perform the avodah together and the pasul person is מתעסק the avodah would be kosher because his inattention to what he is doing neutralizes him [so to speak] and he becomes a non-entity. So all that is left is the kosher person. In this same vein - when a kosher person and a kotton perform the avodah together, even though there is a pasuk that pasels the kotton, the avodah is kosher. This is because he has been disabled due to his status as a מתעסק and thus the avodah would be valid. His avodah doesn't have a שם עבודה.