Q. Is there any reason Timothy's teachings are not part of the scripture?
Deuteronomy 6 is a directive (law/command) to be obeyed by the people and when their children were old enough, they would receive the same command from God as it was written down (scripture) plus part of the blessing is that the children are blessed when they obey their parents. Means that God had appointed the parents to pass down His word so being a parent is being an authority appointed by God.
But in this last days, the only appointed authority is Christ:
Heb 1:
1In the past God spoke to our ancestors through the prophets at many times and in
various ways,
2but in these last days he has spoken to us by his Son, whom he appointed heir of all things, and through whom also he made the universe.
And after Christ, Christ through his appointees.
There is a specific reason why it has to be God's appointed authority and not just anybody else.