Tanchuma at the beginning of our Parsha:
בספרי (פרשת ראה) משמע דאיכא שתי תנופות דקתני והנחתו לפני ה' אלהיך מלמד שטעונין תנופה ב' פעמים אחת בשעת קריאה ואחת בשעת השתחויה
that the bowing was done during the second Tenufa, when the Kohen lofts the basket in the cardinal directions.
We find that the Torah mentions bowing in two contexts. One connotes respect, and the other communicates gratitude (e.g., Breishis 24:51-2, where Eliezer was told that his mission was successful and that he could bring Rivka back to Yitzchak, it says הנה רבקה לפניך קח ולך ותהי אשה לבן אדניך, כאשר דבר ה'. ויהי כאשר שמע עבד אברהם את דבריהם וישתחו ארצה לה' לה).
The halacha (see Brachos 34a) is that in Shmoneh Esrei, we bow at the beginning and the end of two brachos- Avos, the first bracha, which ends with Magen Avraham, and Hoda'ah, which is Modim. These two bowings express both the Gaon and Tosfos' Sifri. When we bow in Avos, as we begin Shmoneh Esrei and reflect upon Hashem's greatness, this reflects the Gaon's bowing, that one must bow in the Beis Hamikdash to demonstrate one's awareness of and subjugation to Hashem. When we bow in Modim, this reflects the bowing the Sifri refers to, a way to express our gratitude for Hashem's love and beneficence. As Rebbi Chanina says in Brachos there, דא"ר חנינא ראשונות דומה לעבד שמסדר שבח לפני רבו..... אחרונות דומה לעבד שקבל פרס מרבו ונפטר והולך לו.
So the Tanchuma becomes crystal clear. When we bring Bikkurim, we bow to express these two feelings: Awe/Humility and Gratitude. Bikkurim is the highest opportunity to express these feelings of humility and gratitude, and Hashem intimated to Moshe Rabbeinu that this experience is vital to our spirituality and ought to be eternal. Moshe Rabbeinu, foreseeing that the Bikkurim would not last forever, instituted the Shmoneh Esrei, which forever formalizes our ability to express these two feelings.
Chazal (Brachos 34b) tell us that bowing during Tefilla should be limited to those places where it is prescribed, and not done wherever one feels like it. As mentioned above, in Shmoneh Esrei it is found only five times- the beginning and end of Avos and Hoda'ah, and when we step back at the end. But most people have the minhag to bow when we say Barchu at an Aliyah to the Torah and when we say Barchu cafter Yishtabach at the beginning of Birkos Krias Shma. Interestingly, when I spoke to my shiur about this, some said "Nobody bows down by the Borchu of the Aliya," and the others said "Everyone bows down the Borchu of an Aliya." But I think bowing by the beginning of Birkos Krias Shma is pretty universal. Where does this minhag come from? Is it a minhag that is brought in the poskim? Why the difference in minhagim?
(Aleinu is entirely unrelated to this discussion, because there, the text states ואנחנו כורעים ומשתחוים, so it would be wrong to say it and not do it, like the עדות שקר of Krias Shma without Tefillin- Brachos 14b).
I was directed to an article by Rabbi Daniel Rakkah in the publication Ohr Yisroel, in which he brings an abundance of sources for kriyos in davening. The article is in the fifth volume of Ohr Yisroel, which as of this writing is not available on Hebrewbooks.org. The Editor/Publisher was kind enough to send me a pdf, which can be accessed here.
Thanks to the wonders of the internet, several of Rabbi Rakkah's marei mekomos are easily accessible. Here are the ones that are currently online:
The one that he brings that I cannot get online is the פסקי מהרי"ץ from רבי יחיא צאלח. The Piskei Maharitz quotes a well known Medrash but with the addition of a few words that are unique to his version of the Medrash and which make it relevant to this discussion.
The usual version of the Medrash is
By the way, I saw the Yalkut Yosef (3:421, 57:9) cited as saying that Rav Ovadiah Yosef does not bend at all by the Barchu of Birkos Krias Shma.