Jesus said "one lord, one faith, one baptism". The problem arises when baptisms are done in His name but without His authority. We believe that "one must be called of God by the laying on of hands by those who have the authority to preach and administer in the ordinances thereof." In other words, the authority to perform God's saving ordinances, including baptism, was lost when the original apostles were killed without their being able to replace them with others who were similarly called of God. The falling away happened and the restoration was needed. That took place in the 19th century when Jesus and the Father appeared to Joseph Smith and began to restore the missing authority. Joseph was ordained to the priesthood of Aaron by John the Baptist, the same that ushered in Christ's ministry. That was followed by Peter, James and John ordaining him to the Melchizedek Priesthood and Elijah granting the sealing powers of the temples. With that authority, we could move forward to not only baptize the living who were converted but also to perform vicarious baptisms for our ancestors who did not have the opportunity to be baptized by authority while in this life. Such baptisms were performed in Christ's day, else Paul would not have mentioned it in his discussion with the Corinthians.
While I understand your reluctance to need another baptism, it is necessary from our point of view in that the proper authority needs to be manifest.