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Koans seem to be crucial in some sects of Zen to achieve 'satori', enlightenment, or liberation, but I don't think this is in the same context as 'mantra', which is a tool for meditative one-pointedness, preferred by some people in various schools. The purpose of a koan in Zen, conversely, seems to be no-pointedness, both literally and figuratively, in that by tricking the mind into a narrative blind alley, the only way out is enlightenment, beyond the absurdity of narratives, including doctrines, which do not accurately represent truth or reality. So yes, this would seem to be a path toward Buddha nature, if not Buddhahood, peculiar to Zen, with beginnings in China, and bought to fruition in Japan. Theravada seems to try to accomplish something similar by avoiding discursive thought all together, whether with or without a mantra, such as 'Buddho Buddho Buddho', etc...

In Chinese Chan Buddhism -starting with master Zonggao (1089-1163)- it was said in matters of “stories” (gongan) that:”If a sentence is understandable, it is not good for meditation; only those sentences without answers are useful.”. Creative process teacher Hermanson writes:
In my Psychology of Creativity courses, I go even further. I want my students to leave their rational minds behind and let their imaginations take over. The only way to do that is to intentionally give them something confusing. One of my favorite and most powerful exercises is to have the students speak gibberish to one another. The mind can’t comprehend what is being said, so they’re forced to step into a “third space,” an “imaginal” space, where a new language comes to them through mental images, story and metaphor. My goal is take them out of their element in whatever way that I can. When we’re confronted with something that our very smart left-brains cannot even begin to figure out—that’s when the magic happens. That’s the moment we have a space for the imaginal. (2013:no page).

The third reason for the shortage of reported schizophrenia cases in ancient times may have been that milder forms of mental illness often went unrecognized by families because such behaviors did not result in any social disturbance or personal distress. This situation is not that old; early in my career, I met several families of psychotic patients that had additional members whose thinking was clearly disordered. The families typically framed those members’ nonsensical language as poetic, mystical or esoteric. -...-Divination practices were almost always conducted by shamans, and consequently, each ritual became imbued with supernatural trappings. In fact, it may have been the disordered thinking of the shaman that helped inject randomness into such affairs. -...-Cults are formed when three stars align: 1 — a charismatic personality endowed with 2 — mildly delusion ideas represents 3 — a marginalized segment of society. Parenthetically, a good proportion of this marginalized segment may be excluded from mainstream society precisely because they exhibit small measures of psychosis (especially idiosyncratic or disordered thinking). In large, complex societies, this triad is the fountainhead of new religions. Such a potent combination could have occasionally been present in traditional societies; however, the typical extremism so often seen in cults would have normally been tempered by skeptics in the tribe. Accordingly, anthropological accounts show that although shamans were customarily influential in group-level decisions, they were by no means the final word.(Polimeni 2012:25,27,38,145).

Mantras are codified prayers and/or invocations (such as NAMO AMITUOFO) and have little to do with KOANS (what's the sound of one hand clapping?).

FWIW