The "Liturgy of St. Tikhon" is not a liturgy of St. Tikhon at all, nor was it ever approved by the Russian Synod of Bishops at the behest of St. Tikhon. As explained by the Antiochian Archdiocese:
"In 1904, Archbishop Tikhon received a response from the Holy Synod of the Russian Orthodox Church to his inquiry regarding the potential Orthodox use of the "Holy Communion" from the American Book of Common Prayer. The Holy Synod noted various problems, mostly the omission of standard Orthodox devotions, such as the invocation of Saints, and an explicit "descending" Invocation of the Holy Spirit. Archbishop Tikhon was directed to make such corrections as he thought convenient and provide a usable adaptation of this Liturgy for practical use with convert Anglicans. In the three years remaining before 1907 when he was recalled to service in Russia, the Archbishop did not finish this work. Some writers have accused him of failure in that he did not, in his short time in the American mission, produce the corrected Western Rite. The corrected rite from the American BCP was produced seventy years later [at Incarnation Church, Detroit] by the Antiochian Archdiocese and is happily used by a growing number of convert Anglican parishes."
This explanation comes not from an anti-Western Rite source, but from the very group that today uses the “Liturgy of St. Tikhon.”