You have pointed to the source of Atipha as the Tosephta Berachot 6:15. This Tosephta simply states the bracha. I would like to understand where the *idea* of atipha comes from. And equally, the mishnah berurah quotes an earlier source for wrapping the whole tallit around the head. Does this mean that it was not required and therefore not included in the Sulchan Aruch itself, or the opposite, that it was such a common practice in the times of R. Yosef Karo that it did not need to be written?

ב"ה Shalom Sorry for the delay, but your question is fully justified. Indeed the Tosefta just says what the version of the Bracha is without providing background as to why the word "hitatef" was chosen. On the other hand, the Torah does say that "you shall make ="Gedilim" = Tzizit on the four corners of your garment with which you COVER yourself" (Devarim 22:12). So, the Torah itself gives an indication that the body is covered and wrapped as we do with the Tallit. Furthermore, The Bet Yoseph 8, quotes the Geonim who say that a person should enshroud himself like the Yishmaelites , who wrap their heads on all sides.(See Gemara Moed Katan 24a) The author of the Itur (The 12th century Rabbi Yitzchak Abba Mari of Marseille) says that not necessarily wrapping of the head but of the body. But without getting into the practical side of that dispute, we have learned that "hitatef" refers to the way we are covered with the Tallit. See also Tirgum Yonatan ben Uziel on Bereshit 24:68 on the word ותתכס and she covered herself, he translates to אתעטפת using the word "Hitatef" In regard to the Mishna Berura, I don't know exactly where you are referring to. The Mishna Berura quotes the custom of the Ari z"l , that when first putting on the Tallit , we cover our heads up to our mouth, and then toss the 4 corners over our left shoulder for a short time and then we wear the Tallit normally. This is common practice today. Perhaps the practice before the Ari z"l, was just covering our body skipping covering of the head. (See Piskei Teshuvot 8: 5 for more details of practices from earlier generations) However, what is clear that placing the Tallit just around the neck without covering the body , the Mitzvah is not fulfilled. All the best