Pretty much all arguments for Aramaic primacy are focused on the Gospel accounts, especially the Gospel of Matthew. The scholarly consensus is that the Gospels were all originally composed in Greek, but likely relied at least somewhat on sources (whether written or oral) which were in Aramaic.
Many of the earliest church fathers taught that our Gospel of Matthew was a translation of the Gospel of the Hebrews, which most agreed was written in Aramaic despite its name. (Some said Aramaic in Hebrew script, which is odd considering that by the time of Christ Hebrew was already almost always written using the Aramaic script, with only the Samaritans sticking to Paleo-Hebrew.) Their quotations from the Gospel of the Hebrews though shows that it could not have been a literal or direct translation though, as it started with Jesus's baptism by John the Baptist rather than the nativity and included longer accounts of things like why it is so hard for a rich man to enter the kingdom of heaven. (Jesus denounces the rich man for violating the commandment to love thy neighbor by hoarding his wealth while surrounded by so many fellow Jews who are too poor to support themselves.) I am inclined to believe Matthew was composed in Greek but drew primarily from the Gospel of the Hebrews.
However, it seems highly implausible that the entire new testament was originally written in Aramaic. It makes absolutely no sense for epistles written to churches in Greek speaking cities to be written in Aramaic rather than a language that the recipients would understand. As Greek was the language spoken in all of the churches in Asia Minor to whom Revelation was addressed (and at Patmos where it was written), I cannot see much if any value in reading an English translation from the Aramaic.
It would be a different matter if you actually spoke Aramaic (and not Greek), as that could help you pick up nuances that are not easily conveyed in English grammar. I personally do like my Latin Vulgate.