That's an interesting point. But if we really are trying to say the Shir Shel Yom as a Shirah
on the Tefilla (because the Tefilla is instead of the Korbon), and for that reason we say the Shiroh of Tomid shel Shabbos just after Shacharis shel Shabbos - on Yom Tov we are completely wrong, since we say on Yom Tov the shir of weekdays and not even the Yom Tov Korbon Tomid's shir, See Rashi Rosh Hashono (Daf ל Omud ב, and for that reason the Chassam Sofer was opposed to saying on Yom Tov the words "שבו הלויים היו אומרים בבית המקדש ").
Are there more details about the Rov, and in which shul in Frankfurt this happened? In
Divrei Kehillos (Minhogei Frankfurt year 1818, page 76) it is mentioned that Mizmor Shel Yom Hashabbos is said after Musaf (and after Shir Hayichud and Shir Hakovod).
Michael