Is rubbing the teeth with the finger during rinsing an innovation (bid'ah)?

Using the finger for siwak (tooth-cleaning) during madhmadhah (rinsing the mouth) is not an innovation; rather, some scholars have deemed it desirable. The proof for this is what Ahmad narrated in his Musnad from Ali (may Allah be pleased with him) regarding the description of the Prophet's ablution (peace and blessings be upon him), which states: "Then he inserted some of his fingers into his mouth." Scholars have differed on whether the sunnah of siwak is achieved by using the finger. Some said it is only achieved if the finger is rough, while others said it is achieved because the objective is met, and the correct view is that it is achieved. As long as siwak during ablution is legislated, based on the hadith: "Were it not that I would overburden my Ummah, I would have commanded them to use siwak with every ablution," then using the finger for siwak during madhmadhah cannot be considered an innovation.