As far as I can see the only flaw in the confession is the money Walt payed for medical treatment. Why would a drug kingpin need his employee brother in law to pay a couple hundred thousand in medical fees?
Deniability. Think about it. DEA agent on a modest salary is paying for almost $200K in top-notch rehab? I don't think so. Instead, use a cover story that your gambling addict brother-in-law paid for it with his winnings, instead. The best part of said cover story is that it's (mostly) true! This puts all the risk on Walt, which fits perfectly with the narrative of Hank blackmailing and threatening him into doing it. The whole thing hinges on Hank being this master manipulator, a more shrewd kingpin than even Gus Fring. His relentless pursuit of Fring, by the way, is cast in an entirely different light once you believe he was using DEA resources to thwart his competitors. This might be a smarter move by Walt than we realize, though it has no evidence to back it up other than the money Walt paid for Hank's treatment--which is, as Hank said, the "nail in the coffin" for him.