The shul you were attending seemed to have been following the custom of the Vilna Gaon.
Rabbi A Grossman has an article entitled The Vilna Gaon’s Psalms for Special Days.
...the Vilna Gaon was faced with a conflict. Like Maimonides, he
believed that the public prayers officially ended with the reader’s
full qaddish, what we call qaddish tithqabbal, and that ideally there
should be no additions thereafter.
As is evident from his other writings, the Vilna Gaon was opposed to
the recitation of any psalms after any of the prayers, but he did
approve of the practice of reciting the Psalm of the Day because of
its educational benefits. Thus, although it would make sense for us
this coming Thursday (Rosh Chodesh) to recite the psalms for both
Thursday and the New Moon, the Vilna Gaon limited the recitation to
the psalm that would have the greatest impression on the populace.
....since Temple times the sages assumed that the psalm for the New
Moon precedes all the other psalms, even that of the Sabbath, because
it would anounce the beginning of the new month, as the beginning of
the month was something only determined that very day.
As such, the psalm for the New Moon would be the psalm of choice if we
could only recite one during our prayers, whether Sabbath or weekday.
THe article does not source the statements of the Vilna Gaon.