Kol hakavod, Hummingbird, for saying the entire Tehillim on Shabbos for so many years!
I’m not clear myself why we mention cholim on Shabbos by Mi Shebayrach but not by Tehillim (I understand the Torah is present, but still). I hope someone here may know.
Some rabbanim say it’s ok to say names L’iluy nishmas (not cholim) while others say one may just THINK the names.
At the end of Tehillim, there’s a special Yehi Ratzon to say after completing any or all sfarim of Tehillim ON SHABBOS which I’m sure you’ve said many times considering your dedication.
So I think the statement “litvak Ashkenazim do not say tehillim on Shabbos for two reasons” would refer to saying Tehillim prefaced by cholim’s names. Certainly it’s admirable and desirable to say Tehillim on Shabbos. In fact, there are quite a number of Tehillim groups meeting regularly on Shabbos to complete all 5 sfarim.