I think the main problem that psychologists have with the MBTI is this:
The fact the MBTI shows correlation with their preferred test of the same thing, aka the Big5, not withstanding. This correlation means that MBTI is just a different mapping of the same underlying structure.
You tell me what your MBTI types is with percentages, and I'll tell you what your Big5 score is.
Now it can be argued that the theoretical basis of MBTI is outdated or vague, but using that as a main critique when the types themselves seem to be quite descriptive of people just means that some people are unable to tell the difference between the literal and figurative. "Nnnnngg... description as used does not match Webster's dictionary. Brain hurts. Can't think. Reality conflict."
The critique that the discriptions match any personality either shows that people aren't being honest with themselves or that they simply don't know themselves very well. I can't see why the Big5 wouldn't have exactly the same problems, namely because the testing is quite similar.
Overall, I'm pragmatic when it comes to tools. Does the MBTI work? Yes. Is it useful? Yes. Is it some god-given dogmatic truth about the world? No.
I think of the MBTI as a handy taxonomy that can quickly sort people into descriptions that are rather solid. Now a description/classification is not a theory (<- I'm using the scientific definition of theory here. Not the layman). However, I frankly doubt that psychologists posses much of a theory of psychology either given that much of their research is based on the behavior of small groups of western cultured undergraduate who are paid $5 for an hour of their time, neatly summarized with archaic t-tests of the 19th century.