The arguments here seems odd. For those who agree with Reb Moshe but are still machmir on CY, it's indeed just a chumra. Those who disagreed with Reb Moshe's chiddush didn't see any difference from CS to Cholov Akum, no chumra whatsoever. One may say that the vast majority of American Litvaks view Reb Moshe's psak as an ikkar ha'din, but no one can argue that the
vast majority of today's Lakewood consists of Litvaks.
BTW, the vast majority of Lakewood holds of CY as a chumra, not ikkar halacha, unlike Chabad, which may explain this hechshers stance.
It's not just a Chabad thing.