The following is a quick translation of a precious response from Imam Taqi al-Din al-Subki (May Allah have mercy on him) that I come across recently while reading his compilation of ethical-legal responsa (fatawa), known as Fatawa al-Subki. Imam al-Subki said

Note (fa'ida): You asked on Saturday,Rabi` II 752 [A.H.] about the reason (`illa) for the prohibition of recitation of the Qur'an while bowing and prostrating. I said: Prayer encompasses
1) directing oneself [towards Allah] (tawajjuh), which is the aim (maqsud) of the one who is praying [as indicated by his] saying "I turned my face...";
2) recitation of the Qur'an, which is an [act of] devotion (`ibadah) intended in and of itself (maqsudah fi-nafsiha) and
3) standing, which is not an [act of] devotion per se since it is a conventional [posture] (mu`tadah). Rather directing oneself [towards Allah] and recitation of the Qur'an are what is intended [during this posture].

Then when he finishes [standing] he bows, humbling himself before Allah. [Since] bowing is an [act of] devotion intended in and of itself, he does not recite Qur'an during it because one [act of] devotion is not [to be] intruded upon by [another] act of devotion [li-anna `ibadatan la-tadkhulu `alayha `ibadah]. The supplication [that is said] while [one is bowing] is but a confirmation of the meaning of [this posture, as indicated] by his saying: "To you I bowed...". Prostration is similar [to bowing in this] and is even more complete [as a posture in which one] humbles [himself to Allah]. The supplication [recited during it], that is, "My face prostrated..." is a confirmation of the meaning of [this posture.] These three [acts of] devotion--recitation , bowing and prostrating--are the aims of prayer [maqasidu l-salah] that are, in their entirety, sheer directing oneself [towards Allah]. When he finishes [praying] he closes with the Testification (tashahhud) while sitting, which is like standing in that both are conventional [postures], the [sitting] posture not being a posture of devotion [per se]. Praising Allah and then invoking peace upon "His righteous slaves" are performed during [this posture], as though the one praying were detached from this world, advancing upon Allah (Flawless is He!) with the entirety of his self, such that when he is about to return to his senses and to the fellow members of his species, he greets "Allah's righteous slaves" with [the phrase] "peace [be upon you]" and then closes the prayers with [the same greeting], since when one who has been absent comes upon [a group of people] he says "peace [be upon you]." [So], since he was absent during his prayer and has [at this point] finished, he greets the righteous--the angels, jinn and human beings--in his presence.

(Fatawa al-Subki , K. al-Salah, s.v. "Qira'at al-Qur'an fi-l-Ruku` wa-l-Sujud")