Why do we pray Dhuhr and Asr silently, despite the Quranic command: "And do not recite [too] loudly in your prayer or [too] quietly but seek between that an [intermediate] way"?

The choice to read silently in the Dhuhr and Asr prayers is due to people's worldly engagements at those times. If the Imam were to recite aloud, the congregants would become distracted, and some might disturb others if they were praying in a group. As for reciting aloud in night prayers, that time is typically free, allowing the congregants to benefit from the Imam's recitation. Regarding the verse, "And do not recite [your prayer] too loudly or [too quietly], but seek between that an [intermediate] way" [Al-Isra: 110], it pertains specifically to night prayers and Tahajjud, so that the worshipper does not disturb those sleeping with his loud recitation, nor deprive listeners of benefit with his silent recitation. Furthermore, the Prophet used to recite the Quran aloud, and the polytheists would insult him, so he was forbidden from doing so. Then he recited silently, and his family missed out, so he began to recite slightly aloud.