That is an exceedingly optimistic hypothesis.

The people who are competent at math are probably N types on the MBTI, people who are good at abstract reasoning, but who might be less competent at focusing on empirical data and specific concrete situations.

Might be, indeed. This hasn't stopped physics, chemistry, engineering, biology, astronomy, etc. all of which have empirical data and concrete situations, and are chock-a-block with maths.

The sciences, especially the social sciences, need people who are good at observing/collecting data

Indeed they need such people. If you have evidence that the present selection procedures prevalent in the social sciences select for such people, I would be delighted to hear it.

Observing and collecting data is stereotypically something that maths types are good at. Consider google, data science and data mining.

Let me refer to Why is machine learning not used in medical diagnosis?

The top voted answer in the Quora discussion is from a medical student (...) describing the complexity of a diagnostic decision, and claims “the human body is incapable of being defined by any algorithm, no matter how bloody brilliant it is.”

The expert systems in question supposedly outperform human doctors!

The problem with machine learning in medicine is not the machine learning. Machine learning and AI have come a long way since the 80′s, and even then automated systems outperformed doctors in experimental settings.

I hypothesise as follows : the non-mathy fields maintain a group dynamic that causes a certain hostility towards mathematical ideas, even when such ideas are objectively superior. To an extent, this also prevents objective judgement of people's abilities within the field, and steers these fields away from a desirable meritocratic state. We end up with fields that select against mathematical ability (those with mathematical ability flee as soon as they realise that the entire history curriculum does not contain a single course on radiometric dating - I wish I were kidding), and that may not select for other desirable qualities instead.