is there some reason this would imply a flaw in Myers-Briggs rather than a flaw in the other instruments or simply that they're measuring different things?

This is the usual modus tollens/ponens question: just pointing out inconsistency (low correlation) doesn't tell you who to favor. In this case, the argument for rejecting MBTI rather than the others would go something like 'it has a highly questionable origin and does not seem to measure anything interesting; the other scales have good theoretical justifications in their areas or were derived directly from the data, have demonstrated various forms of usefulness like predicting relevant behavior, and are less likely to be collectively wrong than MBTI uniquely correct'

As for measuring different things, well, then you get into other things like the lower psychometric reliability of MBTI compared to Big Five - if a instrument is not reliable, then it may be measuring nothing of interest.