Apparently there are some people who do not rely on the OU for kashrus. I have seen a few online references (i.e. this Matzav post) that claim this makes no logical sense, since virtually every other hechsher relies on the OU for ingredients. Is this true?
It's not true that the claim makes no logical sense. Although most or all other kashrut orgs rely on the OU to some extent, they do so to varying levels: Some will rely on it totally, others will rely on it only for uncomplicated ingredients, or only in cases where there is no possibility of conflicting standards. I believe the OU itself has an internal policy concerning which orgs are fully trusted and which are relied upon only in limited cases.
I do not have the answer to your question but I do know that in order to answer this question you must know all of the kashrus related responsibilities the OU fulfills and which of them are taken over by the other hechsher. The ingredients of an item is not the only criterion required to make something kosher. I think that this question might be assuming an oversimplified view of the enormous system required to ensure that something comes out kosher from start to finish.
Just to make it clear, I am embarrassingly under-educated in this arena and would love to hear people's input. I am just cautions of people seeing one piece of the entire system, taking it out of context and judging it based on a position that is both ignorant of the halachos and the system that any given agency has in place.