All perfect praise be to Allaah, The Lord of the Worlds. I testify that there is none worthy of worship except Allaah, and that Muhammad is His slave and Messenger.
The books which you mentioned, dear brother, are books of the Hanbali School, and the scholars of this School are of the view that it is an obligation to rinse the mouth in ablution and in Ghusl (ritual bath). As regards what you mentioned about their view regarding removing solid things between the teeth, we did not find this in their books, but they have stated that the obligation is achieved by the slightest movement of water around inside the mouth.
The view we adopt here at Islamweb is the view of the majority of the scholars that rinsing the mouth and nose is not an obligation in ablution and Ghusl because the Prophet, sallallaahu ‘alayhi wa sallam, ordered in Ghusl to pour water over the external parts of the body, and he said to Umm Salamah as in the Saheeh of Muslim: “It is enough for you to pour water on your head three times and then pour water on the rest of your body, and in this manner, you become pure.”
The inside of the mouth and inside of the nose are not from the external parts of the body, so command to wash does not include them.
Allaah Knows best.