As far as I understand it, that is mistaken on two counts:
First, because the Mormons believe that the Church apostasized after the death of the apostles and became the “Church of the Devil.” Mormonism then claims that the true church was restored through the Prophet Joseph Smith in the 1820s. Their belief that the Church was utterly lost in damnable error for some 1500 years would make a claim of unbroken apostolic succession impossible (some Mormon correct me here if I've got that wrong);
Second, the Catholic church argues that the Catholic and Orthodox churches --- which together would account for approx. 70% of the world's Christians--- have unbroken lines of apostolic succession.
Hence, it follows that most Christian churches have an unbroken line of succession..
“First, because the Mormons believe that the Church apostasized after the death of the apostles and became the “Church of the Devil.” Mormonism then claims that the true church was restored through the Prophet Joseph Smith in the 1820s. Their belief that the Church was utterly lost in damnable error for some 1500 years would make a claim of unbroken apostolic succession impossible (some Mormon correct me here if I’ve got that wrong);”
You are close. We do ascribe to a concept we call the “Great Apostacy,” and what became known as the Catholic Church was, indeed, part of this apostacy.
We do believe that the apostolic authority, lost during the Apostacy, was restored through Joseph Smith by Peter, James, and John.
We do -NOT- believe that the Catholic Church ever became the “church of the devil.” That is an opinion held by some old-fashioned Mormons, but it is not LDS doctrine. In LDS terms, the “church of the devil” refers to those who work to oppose the sincere followers of Christ - and both groups transcend ecumenical barriers.
Best regards.