Add to that "People receive different MBTI results even in the same week."
If I ask you what your weight and height is today, it's indicative of what your weight and height will be a year from now even if the number doesn't fit into exactly the same rounded range. Doesn't mean weight and height are meaningless if the answers change classifications a bit.
A much better analogy would be blood sugar level. For many healthy people its sorta constant-ish. The fact that its not nearly constant for a subset of people doesn't prove the test is therefore worthless. It just proves some people have untreated diabetes. The test is not worthless because it can be used in diabetes screenings, unless the point of the test was to tap people like maple syrup trees or something really weird like that.
A major problem with the test is we know people are not terribly good at self introspection, we're all above average drivers and lovers and thinkers, all of us. I am good enough at self introspection to tell I'm pretty well introverted, yet I can imagine someone not so observant or self aware or perhaps neurotic about the whole topic, getting a fairly random score. Or I'm bad enough at self introspection to not realize I'm no good at it at all, and I'm actually a closet extrovert, although I find that rather unlikely (although I would, wouldn't I, if I failed at introspection).
In that way as a secondary measurement I'd propose theoretically that the wider the variation in test results, the worse the person would score in some kind of self introspection ability test. It seems a reasonable hypothesis. You'd need correction factors to deal with people having diagnosed or undiagnosed swings in brain neurotransmitters or use of addictive substances (including sugar and caffeine). But it could be run as a study.
That said there are other criticisms, for instance: Are thinking and feeling really an axis or are they two independent traits? Are there other useful axes not included?
While I sort of cringe at the idea of using this for assigning people jobs, there area few people in my life that fit some of the type narratives fairly well and it helped me understand their motivations a bit better, which did legitimately help my relationships with them.
That doesn't seem like the proper question. Specifically, it works only to predict future MBTI results, not anything further. That's fine for height and weight because those values are robust to small/moderate changes in value.
MBTI already has low predictive power about real-world content, and high sensitivity to change (on the margins; it also suffers from applying a clustered model to non-clustered data). So if retest reliability is anything less than excellent, I worry that our predictive power has been almost totally destroyed.
To provide a new analogy, this is like variance in number of fingers. The significance of 11 vs 10 vs 9 isn't proportional to magnitude, so we wouldn't accept "small" counting errors.
None of that says that "I'm an INTP" is meaningless, but I suspect that MBTI is no better, or even worse, than non-formalized personality summaries.
(Of course, MTBI is neither accurate nor precise. It is not accurate because it does not provide us any useful information about personality. It is not precise because test results vary wildly.)