The issue is that the mind is by nature subjective and wholly private, inaccessible to outside observers. Further, whereas when conducting an experiment in physics where it becomes possible to separate the experimenter from the experiment (to an extent, observer effect and quantum mechanics notwithstanding), as regards matters of the mind the observer and the observed are one and the same of necessity; you are in fact part of the experiment yourself.
> The problem with psychology experiments is that the mind has many hidden variables which cannot be easily accounted for.
This is where the pillar of quantifiability breaks down, and barring advances in techniques for inspecting the brain with greater spatial and temporal resolution it's hard to see how one can quantify what cannot be directly observed.
Empiricism is the only aspect of the three aforementioned that can still hold up within this domain, and there is a large centuries' long tradition of studying the mind subjectively, qualitatively (as opposed to quantitatively), and yet in a still empirical fashion with falsifiable hypotheses; various mystical and meditative practices exist in this space, such as the fire kasina practice, especially as formulated by a trauma physician [0] complete with falsifiable hypotheses and steps for independent reproduction. What distinguishes practices in this space from harder sciences is that the phenomena observed are subjective, not "out in the physical world," and there does not exist any instrumentation for measuring them apart from that of your own faculties of perception.
As an aside I found the comparison to shamans and spiritual teachers in TFA interesting, since I have always considered psychology and spirituality to concern themselves with the same subject matter and problem domain of minds; one could say that they are both proto-sciences of mind, and psychology is the latest iteration of this tradition (though as TFA states somewhat misguided and out of touch with its roots in mysticism; on this latter point see Jung's Psychology and Alchemy for a discussion of where analytical psychology connects with the older esoteric traditions).
[0] https://firekasina.org/wp-content/uploads/2017/11/the-fire-k...