Street language would likely have been Aramaic, Hebrew, or Greek depending on the group of people He was with. We do know that all three were used at the time including Latin although less common. We also know however that Greek was the predominate language of the entire area. Alexander had conquered that region over 300 years before the birth of Jesus. We know that Koine Greek was the common street language of Rome, Alexandria, Athens, and Jerusalem from 330BC to 330AD. It was surely the language of commerce. G.L. Archer wrote this: "Greek was the most ideally adapted linguistic medium for the World-Wide communication of the Gospel in the entire region of the eastern Mediterranean, Egypt and the Near East. Accurate in expression, beautiful in sound, and capable of great rhetorical force, it furnished an ideal vehicle for the proclamation of Gods message to man, transcending Semitic barriers and reaching out to all the Gentile races. [Archer, Gleason L. 1975. Zondervan Pictorial Encyclopedia of the Bible. Vol. 3. Merrill C. Tenney, ed. Grand Rapids, MI: Zondervan.]
Given all that the fact still remains that the Holy Spirit inspired the New Testament to be written and preserved in the Greek language. It is the meaning of the words in the Greek language that we need to do our best to understand the meaning of. The Holy Spirit didn't have to guess as to what Jesus meant when He spoke. Whether Jesus was speaking in Hebrew, Aramaic, Greek, or even Latin is immaterial to this discussion. It is the Greek language that the Holy Spirit had those words recorded in for all time. It is the Greek that we need to understand. Injecting the Aramaic is second guessing what the Holy Spirit had written.
In general, I agree. It's just that in this case we have the Scriptural cross-references ("Cephas") that very much takes this out of the realm of speculation as to the underlying Aramaic.