Saying “there is no reason Rav Moshe would ‘not’ eat from them is not the same as saying he did.
For example; unless you are claiming that Rav Moshe did not keep pas Yisroael, and Bishul Yisroeal, then some of them like OU, are not heschers he could have eaten from (at least not those items where such things would be issues; Like any cereals and breads or cookies etc…) .
Same thing for meat, I do not know that the Rav would eat meat of any and all, of those heschers.
I do not know that he wouldn’t have, but we cannot derive halacha or how a Rav would pasken, based on what people might ‘assume’ he ‘supposedly might have’, done.