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The M"B in Siman 108 Sif-Katan 4 brings that someone who missed a tefillah and is now the shliach tzibbur may use the chazaras ha'shatz for his tashlumim.
(Makor in Radbaz Siman 361.)
The Radbaz does clearly say this. But why? Consider a case where the person missed shachris and is now davening mincha. The shliach tziibur is a shliach to daven mincha for the tzibbur how can he also have kavana (intention) for the tefillah to be good for his teshlumim from Shachris?
The tashlumin that one davens for missing a tefillah is not going back in time to that missed tefillah. When you daven two minchas, one is a mincha, and the other is a mincha that is making up for a missed tefillah (as opposed to the other being an out-of-place shacharis).
This is seen from the halacha that if you miss mincha on erev Shabbos, for example, you would daven two tefillos on Friday night, both of which will be maariv of Shabbos (Shulchan Aruch O.C. 108:9) - even though you missed a 19-blessing weekday mincha, you "make it up" with a 7-blessing Shabbos davening. In fact, some opinions hold (Ta"z) that if you would mess up and accidentally daven a weekday davening, it wouldn't count.
The only difference between the two davenings is that the second one is the one being done because you missed the first. But the two are interchangeable in terms of which tefillah you are davening. In fact, the M.B. cites a halacha that if you davened twice to make up for a missed tefillah, and then you realize that you messed up your first davening of the two, if you did not explicitly intend for the second to be the make-up, the second one will count as your obligatory davening, even though if you hadn't messed up the first would have been the obligatory and the second would have been the make-up. (In other words, the same davening that would have fulfilled your make-up requirement can become your obligatory davening - it seems that one is not mutually exclusive of the other.)
I think the Radvaz answers this with these words:
והשנייה בקול רם יכוין לתשלומין ולהוציא את מי שאינו בקי
With the second out loud he should intend as a make-up and to exempt the one who isn't fluent
The Shliach Tzibbur is being Yotzei the ones who can't say their own Tefilah under the principel of יצא מוצאי - someone who already fulfilled their obligation can say it for and exempt someone else. So when the Shliach Tzibbur is saying Shmona Esre the second time, he generally is doing it to be Motzei others, despite his not having an obligation. He can, at the same time, have a second intention - to fulfill his own Tashlumin. This is because the Tashlumin is a fixing of his mistakes, but is not considered a prayer in its time, rather is a request for mercy, which does not contradict having someone else in mind.
