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?