Do all the oldest Christian sects celebrate mass as a sacrifice of God to Himself?

I read in "The Latin Mass Explained" by Msgr. George J. Moorman that all known ancient Christian sects, distinct from the Catholic church, celebrate the mass as a sacrifice just like the Catholic church. This apparently is the entire point of the mass, that it is a continuation of the Judaic temple sacrifice, but since God doesn't need any created thing we can possibly sacrifice, the only sufficient sacrifice is God Himself, i.e. the doctrine of transubstantiation in the eucharist. In fact, Jesus carries out this sacrifice of himself during the last supper, even before his crucifixion. The idea is that sacrifice is not originally about atonement for sins, since sacrifice existed before the Mosaic sacrificial system. Instead, the fundamental reason is man destroying his most valuable possessions for the sake of God, indicating that all he is, is subjugated to God as the highest thing. However, nothing created equals God, so nothing is truly a sufficient sacrifice except God Himself, and so that is what is sacrificed in the mass via transubstantiation. The book has the Vatican's imprimatur and is written by a monsignor, so seems fairly trustworthy regarding official Catholic teaching on the matter. Is this book's claim correct, that all known ancient churches have the same understanding of the mass, that it is the actual sacrifice of God to Himself?