The biggest problem with cognitivism is that it is very useful. Cognitivism offers a clear framework, and that framework is flexible, in that it allows many different theoretical constructs to be plugged into the same holes; it is utilitarian, in that almost no matter what you put in the holes you can then do something with the product; and it is conformist, in that it is a logical continuation of the Western thought that preceded it. Further (or perhaps as a result), though cognitive terms are used in hopelessly ambiguous ways, they provide a firm illusion of concreteness. The combination of flexibility, usefulness, intuitiveness, and seeming concreteness make it is easy for aspiring psychologists to accept the cognitive approach, and once they do so they can easily go about having a career—performing experiments, publishing papers, getting funding, presenting their work to the public, and interacting with colleagues.
For example, if you were interested in “how memory affects perceived meaning”, how would you go about investigating it? “Memory” could be operationalized in a wide variety of ways; memory could refer to what people remember in the present, any change in behavior following an event you (the investigator) are sure happened, the effects of bodily (neuronal?) alterations on future behavior, the retention of a conditioned response, etc. “Perception” and “meaning” are similarly, if not more, ambiguous. Perhaps you will ask people to recall certain events in their lives, then have them interpret ink-blots; perhaps you will flash lists of words for 50 ms at a time, then see if they feel positively towards words you repeated several times; perhaps you will ask people the perspective from which they view specific memories, then test them for their accuracy in recalling important aspects of the event; the variety is almost infinite. Despite this incredible lack of specificity of the question, it is easily transformed into a concrete empirical endeavor. Whatever the form of the study, the results could be reported in the local newspaper and the average high school student reading it will nod their head as if they are learning something important about human nature. Further, if you also give the task to chimps or dolphins it can make national headlines — “Chimps perception of meaning less affected by memory manipulations than teenagers”— despite the headline saying nothing concrete, it is perceived as understandable and straightforward.
Needless to say, the problem with alternative approaches is that they are much less useful in the ways listed. More on that later, or at the link below.