the thing is that the claim "men are dangerous" is only "bioessentialist" if we also claim that the category of "man" is biologically determined, i.e. that there is some quality of the physical body ("bio") that contains within it the quality ("essence") of man-ness.
if we define the idea of "man" in a way that instead references anything else (as a class defined by a particular relationship to the performance of reproductive labour under capitalism, as a group of colonial ideas that create a certain role within a gender binary, as one possibility for self-identification, literally anything else other than the 'biological' body) then it is not bioessentialist, just because of the definition of what "bioessentialist" means.
the claim "the patriarchy allows dangerous men to exist unchecked" entails the idea that "the category of 'man' exists prior to the existence of patriarchy"—that is, "man" and "woman" are "real" things, "real" preëxisting categories, and then patriarchy afterwards comes along and unethically privileges one over the other. but this—the idea that "man" has a real definition that exists prior to the colonial gender binary / cisheteropatriarchy, rather than being a category or a set of ideals and ideas that was created by the same—is actually the essentialist idea here.
it might not be bioessentialist—you could, for example, claim that the "essence" that creates "man"-ness is spiritual or within the mind, rather than being a quality of the body—but it is definitely and unavoidably essentialist.
the claim that "the category of 'man' is a dangerous one to be begin with" could be essentialist (again, if we make the transphobic claim that a "man" is "a person who possesses a certain kind of body"), but it is the only claim of these two that has the potential to be anything else (i.e., if we define "man" as a particular role created by patriarchy).
effectively, you've got what is and is not essentialist exactly backwards.
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