Mincha Motivation

The Mishna in Yoma (28a) tells of a story that once in the Beis Hamikdash they assumed that it was already considered halachicly daytime. Therefore, they slaughtered the animal designated for the Korban Tamid. They then realized that they had mistaken moonlight for sunlight. The result was that the Korban Tamid was invalid and had to be burned, something that is severely frowned upon. Therefore, they developed a system to ensure that this mistake wouldn’t occur again.
The Siach Yitzchok asks: why didn’t they just wait until sunrise to shecht the Korban Tamid? There is no way that one would mistake moonlight for the light of the full sunrise! Why devise a whole new process, when the solution seems rather simple?
He answers that Bnei Yisrael wanted to perform the mitzvah at the earliest possible moment. They didn’t want to wait until sunrise, if the avoda of shechita was valid before. Therefore, they positioned a lookout on the roof to ascertain the earliest possible time when the Tamid may be brought.
This, the Siach Yitzchok explains, is also the connection to the next passage in the Gemara. Immediately after discussing the practice of the morning Korban Tamid, the Gemara discusses the earliest time for Mincha prayer service. There doesn’t seem to be an obvious connection. However, the Siach Yitzchok explains that once the Gemara discussed the practice of performing the avoda of the morning Tamid at the first possible moment, it discusses davening Mincha at the earliest possible time.