Interesting! This is not the same Rosary… not quite. It’s the same structure and the same mysteries but the prayers are different. I pray this on each bead of each decade:
Joseph, son of David and husband of Mary, we honour you guardian of the Redeemer, and we adore the child you named, Jesus. St Joseph, patron of the universal Church, pray for us that like you we may live totally dedicated to the interests of the Saviour. Amen.
I pray the Our Father and Glory Be as usual between decades.
I conclude it with the litany of St Joseph. I usually throw in Hail Mary’s between the decades too. I also now add a few of the above Joseph prayers between decades of the Marian Rosary - which I pray more regularly.
The St Joseph Rosary at your link assumes St Joseph was assumed bodily into heaven. The version I use doesn’t touch on that… Interesting. Not sure where I stand on that.
This was all outlined on a prayer placard that can be found beneath the statue of St Joseph at my parish.
Why do I find it helpful? Well for one, the mysteries largely parallel the Joyful mysteries, so its another opportunity to reflect on the life of Christ, but from the perspective of St Joseph. And, I simply find St Joseph to be a powerful intercessor!