On 9B we are told that one shita is that the deciding karban for dalus or ashirus is the ashom since it is machshir. In Eruchin 21A the sha"s said that all the korbonus were needed before the mtzorah could eat Kodshim
The Asham is called a Machshir since the sprinkling of its blood on the thumbs of the Metzora etc. is the beginning of the Matir, and the part that involves the body of the Metzora himself (see Rashi here). It is not the entire Matir, as you correctly noted.