Greek primacy "proofs"

[HSB]

I'm not sure if this is the place to ask my question...but it does concern the Greek Septuagint. While I am mostly convinced that the early New Testament writings were composed in Aramaic or possibly Hebrew...why does everybody seem to only quote from the Septuagint when making connections with Tanach. Clearly they had Hebrew scrolls available..yet the Septuagint seems to surface all the time (eg. James/Yacob in Acts 15) Or was the Septuagint considered an exact (as opposed to paraphrased) translation of the Hebrew. In the Peshitta what is used for Tanach quotes...the original Hebrew/Aramaic wordings or the later Greek Septuagint translated backwards. Hope this question makes some sense. Thanks