Is "prayer is better than sleep" in the Fajr adhan an innovation, based on the fact that Bilal used to say it in the adhan for Tahajjud and Ibn Umm Maktum did not mention it in the Fajr adhan, and is comparing it to sleep impermissible, and if so, why is it practiced in Mecca and Medina?

The phrase "Prayer is better than sleep" (الصلاة خير من النوم) is mentioned in the Fajr adhan (definition). This is in the adhan proclaimed after the time for prayer has begun, not in the first adhan that occurs before Fajr. This is because the word "first" (الأول) refers to the first adhan in relation to the iqamah (definition). Authentic hadiths explicitly state that the tathwib is said in "the adhan of the morning prayer" (أذان صلاة الصبح), or "the Fajr adhan" (أذان الفجر), or "the dawn prayer" (صلاة الغداة), all of which indicate that the tathwib occurs after the prayer time has begun.