The gemorah in yuma daf bais is trying to use the word l'chaper to learn that other yomim tovim besides yom kippur require preshia (seperation). I was bothers by the fact that l'chaper (to forgive) doesn't that apply only to yom kippur what does shmene atzeres succoth or shavous have to do with l'chaper why doesn't the gemorah answer this to end all the questions .
Good question. Your point is addressed by a Rishon, RABEINU ELYAKIM, who writes that all of the Yamim Tovim have an element of Kaparah, because the Korban Se'irei ha'Regalim is brought as a Kaparah for Tum'as Mikdash v'Kodshav, as the Gemara says in Shavuos 2a. In fact, the verse even uses the word "l'Chaper" with regard to the Se'ir brought on Rosh Chodesh and the Yamim Tovim (Bamidbar 28:22, 30). That is why the Gemara here has a Havah Amina that "l'Chaper" refers to those days.
Mordecai