selichot - whoever leads s'lichos should lead the rest of the day's prayers: why s'lichos?

Rama 581:1 says whoever leads s'lichos should lead the rest of the day's prayers, and Magen Avraham, followed by others including Mishna B'rura, says this is due to "hamaschil b'mitzva om'rim lo g'mor", "whoever starts fulfilling a command, we tell him to finish", and includes shacharis, mincha, and maariv (although which maariv is unclear; see Machatzis Hashekel). (Incidentally, Aruch Hashulchan notes that this was not the custom in his area at his time. In any event:)

My question is, why s'lichos specifically? Why don't we say, for the same reason of "hamaschil b'mitzva...", that whoever leads shacharis any day should lead the rest of the day's prayers?

Update: I see that the L'vush gives a different reason. [Apparently, in his time and place, a set chazan would lead shacharis every day, and whoever led shacharis on a s'lichos day would fast.] Thus, he writes, have the s'lichos leader lead shacharis also, instead of the usual shacharis leader, [and, I gather, vary the s'lichos leaders from day to day,] in order that no one person need to fast every day.

However, this doesn't answer my question on the Magen Avraham and those who quote him, who don't give that reason.

(Note also that the L'vush's reason would seem not to apply nowadays, when the s'lichos leader does not AFAIK fast (and when we don't necessarily have the same leader daily, although many synagogues do de facto, as there is a mourner leading). But that's another issue.)

Answer

Nit'ei Gavriel (footnote 12 in the link) asks of this practice: What do we mean by המתחיל במצוה אומרים לו גמור? Aren't the following Shacharis and Mincha and (previous and/or following) Ma'ariv seperate mitzvos?

He answers that s'lichos is actually a prelude and preparation for the other t'fillos of the day. For example, if one says s'lichos, but then leaves and neglects to pray shacharis and mincha, he has not fulfilled the obligation of s'lichos either. It must be that the s'lichos are connected somehow to the other t'fillos of the day. (I'm not so sure about the connection to ma'ariv. It's unclear.)

According to this, the answer to your question seems pretty clear. On a regular day, each t'filla is a separate mitzva, and each can have a different ba'al t'filla. But on a day with s'lichos, the s'lichos are connected to the other prayers of the day. (That is, even though shacharis and mincha are seperate from each other, they are both connected to the s'lichos of that day.) Thus, the rule of המתחיל במצוה אומרים לו גמור becomes relevant for each seperate remaining prayer of the day.