According to your view the Christians of today have abandoned the ritual slaughter but I found that even today the Christians in Ethiopia adhere to the mosaic law of slaughter ( The food taboos found in the Old Testament are observed by most people as the Ethiopian Orthodox Church prescribes them. The flesh of animals with uncloven hoofs and those that do not chew their cud are avoided as unclean. It is nearly impossible to get pork. Animals used for food must be slaughtered with the head turned toward the east while the throat is cut: "In the name of the Father, the Son, and the Holy Ghost" if the slaughterer is Christian or "In the name of Allah the Merciful" if the slaughterer is Muslim.) This article I found on the page http://www.everyculture.com/Cr-Ga/Ethiopia.html. The Christians that adhere to the Mosaic Law in Ethiopia pronounce the name of The Father the son and the Holy Ghost. So my question is can we eat that meat and if not why? If yes then why do we allow a believer to eat the meat of their slaughtered animal when they do not only pronounce the name of Allah but add the name of the son and the Holy Ghost? We see the same critic in the Qur'an: They do blaspheme who say: God is one of three in a Trinity: for there is no god except One God. If they desist not from their word (of blasphemy), verily a grievous penalty will befall the blasphemers among them. (Surah 5:72-3)
If the Christians you are referring to are observing the Sharia of their own for slaughtering animals, as you described, then there is no problem for a Muslim to eat from the meat. You are asking how this can be possible when the Qur'an condemns the belief of the Christians of the time in trinity. The answer is, for the same reason that the Qur'an allowed using the meat produced by the Christians who were condemned. We read in the Qur'an:
"… and the food of the People of Book is permitted for you …" (5:5)
Note that these were the same people who were condemned in the same Sura, verses 72 and 73.
The reason behind this is that while Christians do have act of Shirk in their belief in trinity, in their own understanding they are monotheist and believe in only one God. This makes them categorically different from those who knowingly believe in more than one god. This is exactly the reason why in the Qur'an, while they are severely criticized, they have never been called a Mushrik polytheist). It is for the appreciation of their original monotheist belief that Muslims are allowed to use their meat products when the meat is the product of a Sharia based slaughtering.