I've been meaning to post this for a while, but I've been very busy, and only now have a chance to post.
In discussing Shmuel, and how he paskened in front of his rebbe, the gemara explains that it was in regard to shechita.
Eili held that the kohen had to do both the slaughtering and kabbalah. Shmuel derived from the passuk that anybody is able to do the shechita, but the kohen had to do the kabbalah.
The way Shmuel learned was indeed the correct way of reading the pasuk, so why did Eili hold that the kohen needed to do the shechita as well?
I believe it was the Malbim who said the following pshat: In general, a person is forbidden to wear shatnez. The only exception were the kohanim, who wore the avnait - belt - which was made of shatnez. However, they could only wear this avnait if they were doing the actual avodah. If they weren't doing it, it had to come off. So if a regular person is doing the shechita, the kohen who is catching the blood ends up not doing anything during the shechita, so he can't wear the avnait. By the time the person does the shechita, the kohen doesn't have enough time to put on the belt and catch the blood. Therefore, Eili held that the kohen also had to do the shechita, so that he can be there to catch the blood.