Short answer: there is no new, accepted “central insight” that suddenly turns the Wang–Xu 61‑stem computation into a proof that a Kervaire invariant one class exists in dimension 126. The existence in 126 is still open. The 8‑page preprint you cite does not contain a vetted argument; it relies on a step that is exactly the piece of 2‑primary information Wang and Xu left unresolved because it is equivalent to deciding the 126‑dimensional Kervaire problem.

What experts have known for a long time (and what Wang–Xu make completely explicit) is the following.

• The would‑be Kervaire invariant class θ6 in π126^S is detected at the prime 2 by the Adams E2 class h6^2. The only plausible obstruction to its survival is a specific d3 differential in the Adams spectral sequence: d3(h6^2) ∈ E3^{5,130} which would land in the 125‑stem. By standard relations (Toda brackets/May’s higher Leibniz rule/multiplicativity), that potential target can be expressed in terms of σ times a class coming from the 61‑stem.

• Consequently, deciding whether h6^2 survives is equivalent to a very concrete divisibility/extension question in π61^S: roughly, whether a particular class x in π61^S is divisible by 2 (equivalently, whether a certain hidden 2‑extension holds in the Adams E∞ term). This reformulation is classical; versions go back to Barratt–Jones–Mahowald’s relations among the Kervaire elements and to Kahn–Priddy transfer considerations. A convenient way to phrase it is: θ6 exists if and only if a specific 61‑stem element has order 4 rather than order 2.

• Wang and Xu’s 2017 computation of the 61‑stem is extremely precise, but they deliberately leave exactly that hidden 2‑extension undecided, precisely because settling it is tantamount to settling the Kervaire problem in dimension 126. Their differentials are consistent with either outcome for that extension. This is why their paper does not claim a resolution of the Kervaire problem.

• The Minami manuscript tries to fix the outcome of that extension (and thereby force the survival of h6^2) from Wang–Xu’s data, but the argument does not withstand scrutiny: the critical step is not justified. In particular, it treats as determined what Wang–Xu explicitly identify as the unresolved 2‑extension/indeterminacy that is equivalent to the Kervaire question. Experts quickly pointed this out, and the community has not accepted the claimed deduction.

So, to the spirit of your question “what is the new insight?”: the reduction of θ6 to a very specific statement about the 61‑stem is not new—this reduction is the standard “flavor” behind why the 61‑stem is the critical range for the last possible Kervaire class. What remains missing is exactly the extra input needed to decide that hidden 2‑extension. Hill–Hopkins–Ravenel’s equivariant technology kills all higher candidates (j ≥ 7) but is silent in dimension 126, and the currently known Adams/May and transfer/Toda‑bracket relations reduce the 126 problem to that one stubborn 61‑stem phenomenon. No accepted new technique has bridged that last gap yet.