In each negotiation about a position, the company is negotiating only about one of its many positions; the candidate[1] about what will (usually) be their only job. And companies usually, AFAIK, get a lot more applications[1] per position than each applicant[1] gets offers per job search.
So no, AFAICS you are wrong and the original thesis of "the company holds the advantage" is correct.
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[1]: Consider the connotations of those words.
In this case, obviously a job seeker does have many possible employers to apply to.