I understand from a helpful earlier MO question that the techniques leading to the celebrated resolution of the Kervaire invariant one problem in the other candidate dimensions yield no insight on dimension 126, to the extent that until recently there was not consensus as to which way it was likely to go.
For a few weeks now, I've been aware of a manuscript of Minami which, if I'm reading correctly, asserts the existence of such an element in dimension 126 as a consequence of Wang and Xu's 2017 paper on the stable 61-stem. Presumably they would not have failed to remark on the resolution of the Kervaire invariant one problem if such was readily visible from what they had done, so it would seem that there is something that is not obviously a consequence of it, but which can nevertheless be extracted in eight pages if one does somehow know what to look for. It's common that when an old conjecture like this is resolved, it is as an application of some newly discovered techique or heretofore-ungleaned insight into the inner workings of the encompassing theory.
What is the new insight here?
The work on the stable stems is forbiddingly technical work I am unlikely to ever grasp the full details of, but it's still possible sometimes to get the flavor of the ideas involved, and that's what I would hope for here.