Yes the LDS believe it is a pure restoration of first century Christian liturgy and beliefs.
True to the Faith specifies what was lost during the Apostasy:
After His Crucifixion and the deaths of His Apostles, the fulness of the gospel was taken from the earth
About the restoration, the LDS specifically believe:
When Jesus Christ was on the earth, He established His Church among His followers. After His Crucifixion and the deaths of His Apostles, the fulness of the gospel was taken from the earth because of widespread apostasy. Through the Prophet Joseph Smith, our Father in Heaven and His Son Jesus Christ restored the fulness of the gospel. The true Church of Jesus Christ is on the earth again. Because of the Restoration, the teachings and ordinances necessary for salvation are available to all people.
The 'fulness of the gospel' is defined in Mormon Doctrine (not an official LDS source) as:
The phrase "fulness of the gospel" refers to the whole doctrine of redemption demonstrated and taught in the ministry and life of Jesus Christ. It "consists in those laws, doctrines, ordinances, powers, and authorities needed to enable men to gain the fulness of salvation"
This is reinforced in part from Joseph Smith History 1:34 in reference to the Book of Mormon:
He said there was a book deposited, written upon gold plates, giving an account of the former inhabitants of this continent, and the source from whence they sprang. He also said that the fulness of the everlasting Gospel was contained in it, as delivered by the Savior to the ancient inhabitants;
President Ezra Taft Benson explained this by saying:
That does not mean [The Book of Mormon] contains every teaching, every doctrine ever revealed. Rather, it means that in the Book of Mormon we will find the fulness of those doctrines required for our salvation. And they are taught plainly and simply so that even children can learn the ways of salvation and exaltation1
1 Benson, Ezra Taft. A Witness and a Warning.