How is the Prophet's saying, "that describes the voice of David (peace be upon him) as a mizmar (flute/reed pipe)," understood in light of other hadiths that prohibit musical instruments?

There is no contradiction between the prohibition of musical instruments and the permissibility of the mizmar. The prohibition is established by many proofs, such as the Prophet's (peace and blessings be upon him) saying: "There will be among my Ummah people who will consider unlawful sexual intercourse, silk, alcoholic drinks, and musical instruments as lawful," and his saying: "When singing girls and musical instruments become widespread, and alcoholic drinks are consumed." As for the mizmar mentioned in another hadith (definition), what is meant by it is a beautiful voice, as mentioned by Ibn Hajar, Al-Nawawi, and others.