The idea that the Lord’s Prayer (Our Father) is at the beginning for the reason you stated is not historically accurate, still, starting the rosary with an Our Father and three Hail Marys does draw our attention to the three cardinal virtues and our need of them.
The original rosary consisted of praying 150 Hail Mary’s by those who could not recite the 150 Psalms of the Divine Office. It was a way for lay people, who didn’t know how to read or write their own languages, let alone the Latin recited by religious, to join in the daily prayer life of the Church. It gained popularity after Our Lady’s vision to St. Dominic to use the rosary to fight heresies of his day, and it was quite effective. The Our Fathers were a later addition between the decades–when isn’t exactly known–along with other prayers, such as the Glory be. Although most believe the rosary’s main form was set by St. Dominic’s time. After the visions of Fatima, other prayers were added at the end of each decade. The Irish have a sweet expression for such added prayers. They call them “embroideries.”