As someone who has struggled through an essay on characterization in Angela Carter's The Bloody Chamber using Jungian archetypes I may be able to shed some light on the issue.
The archetypal approach, in my experience, is quite controversial and has always been niche (I have never encountered in my studies a contemporary academic or critical work using Jungian archetypes on any text) because of the same reasons that I struggled with when using them in my work.
Firstly, the actual list of Jungian archetypes is quite short and by no means comprehensive- it is almost useless for any characters who are three dimensional, as archetypes prescribe predictable behaviors and qualities that aren't present in the kinds of character driven fiction that have been popularized since the 18th century. For example, I argued that attempt to pin the protagonist of The Bloody Chamber down is entirely reductionist, as even though she is not particularly complex, because she is at times both passive and resistant to control, she does not fit an archetype. It may be evident here how simplistic archetypes are when working with complex characters. As a critic, I am simply not interested in any kind of critical lens that limits my ability to explore different avenues of interpretation, and archetypal criticism does just that, it is my feeling that this is the view of the majority of contemporary critics as well.
Secondly, more in regards to the cultural limitations of Jungian archetypes, the foundation of this lens has been attacked quite successfully (although attacks have come from both literary and psychological sources, Andrew Neher's Jung's Theory of Archetypes: A Critique is a succinct attack on the foundation of archetypes, although I used C.C. Drake's Jung and His Critics primarily in my work). The foundation of archetypal criticism relies on the view that these archetypes are fundamental and transcendental across all cultures-Jung believed they were an inherent part of human psychology. However, there was little evidence for this in the first place as it is quite easy to apply any arbitrary archetypes to almost any myth (a method archetypal critics used to demonstrate the truth of their method) as these stories were not focused on the characters for their own sake, but rather the imparting of supposedly universal truths. However, as critics have noted, Jung draws from a Eurocentric set of archetypes, primarily featured in Norse myth, and to suggest that what is European is fundamental is both culturally problematic and empirically wrong, in cases where these archetypes do not fit myths from non-European traditions.
Thirdly, archetypal criticism tends to become quite prescriptivist, as unless archetypes are just dismissed altogether for characters who are anything other than flat, the attempts to fit characters into archetypes tend to reduce those characters to descriptions that are inadequate or uninteresting. In contemporary criticism, the text precedes the critic, whereas in archetypal critic, the critic precedes the text; meaning that archetypal analyses tend to be an attempt to fit the text into predefined assumptions, rather than looking at the text as it really is and seeing what can be learned from it.
In summary: in my experience archetypal criticism is not wholly useful for round characters, and in many cases flat characters; but I can imagine it may be a point of curiosity when dealing with older texts that are not driven by characters or deliberately allude to archetypes (The Wasteland is a rather famous example of allusion to archetypal figures such as the Sailor). In general, critics want their lens to expand the scope of their analysis, not limit it-and as archetypes tend to be prescriptivist and are from a set list, archetypal criticism limits the lens. In my view, which appears to be the popular view, archetypal criticism is not particularly well supported to begin with, and in the face of complex characters from a variety of cultures, it tends to be an inadequate lens to work with.
I understand that my answer is contentious and in many ways subjective, but as someone currently involved in academia, I can confirm at least in my institution and experience of criticism; this is broadly the view on archetypes. If you would like a more pro-archetypal view, I would recommend reading Jung's Man and His Symbols, the source of Jungian archetypes.
