“but I have davened in other shuls where it is said at a different time. I say the psalm for the day of the week after mussaf on Shabbos and Yom Tov, but I have been to shuls where it is said after shacharis. Knowing that these are possibilities reduces that moment of, “what on earth??” when I encounter such a variation.”
I like your point, but davening with Yitzchok Yair would in no way prepare you for these situations. It only has the shir shel yom after musaf and only has bameh madlikin before maariv.