I think the betrayal of the KR might stem from events after the day of Recreance. We know from Dalinar's vision The KR gave up their shards to a bunch of soldiers or at least two orders did, but who beyond those there would know the new shardbearers were not KR? The new shardbearers could have posed as KR or just let everyone assume they were KR, but not follow their ideals. The new bearers then could take advantage of a populace that relied on the KR for protection and trusted them to act honorably and ethically to gain power, prestige, and wealth. I think one of Jasnah's notes hints at this where it talks about tariffs at Urithiru becoming unreasonable and showing the true nature of the so called Radiants.
Edit: I found the note it is at the beginning of Chapter 46.