The purpose of baptism to be washed from past sin, both original and actual (if relevant to that individual). The Novus Ordo rite of Baptism follows the Novus Ordo theology of baptism is "to join the Church."
No and yes. The purpose of Baptism is based on the first question the priest asks the adult or godparents of the child:
Q1 - "What do you ask of the Church?"
A1 - "Faith".
Q2 - "What does Faith provide?"
A1 - "Everlasting life."
The purpose of Baptism is Faith. It is not the remission of sins, which is a byproduct of the sacrament. You cannot enter heaven without Faith, which is why Christ said that water and the Holy Ghost is required. The Holy Ghost provides supernatural Faith, after the catechumen studies the faith (i.e. religion) and has the necessary (human) faith and desire to enter the Church.
You are correct that the novus ordo's change from Faith to "joining the Church" is a watering-down of the ideal purpose and understanding of Baptism. But it's not invalid.
Thus, again, a BOD'er can desire Baptism (i.e. true Faith), not yet receive it in the sacrament, but can receive remission of sins/justification. But nowhere does Trent teach that justification is equal to the Faith received in the sacrament, i.e. Supernatural Faith/Wedding garment/baptismal character. Maybe it's the same? Not sure. The Church has never clarified this.