This is a nonsensical assertion supported by no serious scholars. The four gospels were obviously composed in Koine Greek. Entire passages depending on Greek construction wouldn't make sense even trying to back-translate them into Aramaic. Same with puns that rely on Greek.

Serious historical critical scholars dispute a lot about the gospels' authorship and textual changes over time — but they don't dispute the obvious fact that they were composed in Greek — as was the Book of Acts, which was written by the same author as the Gospel of Luke. Besides the anonymous author claiming authorship of both Luke and Acts, the Greek style and usage in both books is nearly identical, and very different from the Greek used by the other gospel writers.

As for Paul's letters — they had to be written in Greek given he was writing to Greek speakers in the Levant and Asia Minor who would have had no use for Aramaic texts. Paul also wrote to the early proto-Christians in Rome, but since literate Romans in Rome (a small minority of upper class, educated people) were usually literate in both Latin and Greek, Paul's writing in Greek is to be expected.

Also, Greek was the lingua franca of the Roman empire. Pretty much everyone in the Eastern empire spoke Greek, and literate people in the West did too.

There are some fringe hypotheses claiming Aramaic origins of the Gospels (though not of Paul's letters), but no serious scholars have found evidence to support the fringe ideas, which you report on in an exaggerating manner, grossly misrepresenting scholarship in ways typical of conspiracy theorists.