“No talmudic discussion regards the recitation of [the Song of the Day] as anything other than temple practice”
…the daily Levitical songs (Pss 24, 48, 82, 94, 81, 93, 92) as recorded in a baraita appended to the end of m. Tam. as evidence. According to this baraita, the Levites would sing these psalms during the burning of the daily sacrifice. This passage provides proof for a rabbinic memory of a Second Temple practice, nothing more. There is no evidence that these psalms were part of daily rabbinic liturgy until well after the talmudic period (ca. 650). No talmudic discussion regards the recitation of these psalms as anything other than temple practice. Contrast this with other prayers that are initially imagined as part of temple ritual and are discussed elsewhere in terms of contemporary rabbinic liturgy, such as “True and Firm” (emet veyatziv) (m. Tam. 5:1; t. Ber. 3:6). The most compelling evidence for their absence in early rabbinic liturgy is the fact that even in the Geonic period these psalms appear to be non-liturgical. Seder Rav Amram Gaon strangely claims that one recites the baraita about these psalms and not the psalms themselves, and the prayer book of R. Saʿadyah contains no mention of them. The earliest positive evidence comes from Maimonides’s prayer book, found at the end of his Book of Love, which claims: “And a small part of the nation has a custom to read after all of these supplications the psalm that the Levites would sing in the temple on that day” (section 34). Even in the time of Maimonides, the daily recitation of these psalms was nothing more than a practice of the few. The early rabbis clearly did not adopt the Song of the Levites into daily liturgy.
-Abraham Jacob Berkovitz, “The Life of Psalms in Late Antiquity” (PhD diss., Princeton University, 2018), 165-166.