Purely materialistic explanation:
First of all, for the most of history across the world's cultures, the concept of empirical testing of ideas and beliefs simply didn't exist. Truth came from authoritative tradition, not experimentation. It took the West centuries to come around to the idea testing ideas materially and that largely driven by practical needs such a long range navigation and the need to create new technologies. Egyptian civilization existed for longer than all civilization since combined, nearly 5,000 years and they never developed the idea of empiricism.
(Note that empiricism and rationalism are not the same thing. Neo-platonism was highly rational yet rejected empiricism entirely. )
(The evolution of empiricism requires a certain level of technology, especially uniformity in the production of materials used for measuring instruments, more than any philosophical or theological shift. A careful examination of the time lines of technology and measurements reveals that improvements in empirical power generates the philosophical and theological changes that required to justify the continued use of empiricism. Form follows function, which means function preceeds form. Likewise, justification follows utility which means utility precedes justification.)
Astronomy became the first science because its measurements depended on angles, and devices for measuring angles report the same results independent of the materials they were made of e.g. a wood divisor, a stone divisor or metal divisor, all report the same angle even though environmental conditions of heat and humidity cause each class of materials to alter in different axes and different degrees.)
Secondly, religions don't exist to provide explanations of the material world but to provide psychological mechanisms to enforce greater social cooperation and cohesion. The actual various religious stories don't actually matter to the religions survival as long those stories induce adaptive behaviors in their adherents i.e it doesn’t matter whether an individual refrains from violence because they fear the judgment of Jesus or they fear bad Karma, it merely matters whether the belief in the story makes them choose non-violent interactions with their fellows over non-violence.
As such, empirical events have little impact on faith in the immediate short-term, because their primary adaptive functions is unconnected to actual material events.
Thirdly, across the world's religions, disasters are interpreted as punishments from the "good" side of the religions instead of being the work of "bad" side, the opposite of what one might expect. But, if you view religions as mechanisms for enhancing cooperations, it makes perfect sense, as the response to a disaster would be an increase in cooperative behaviors as individuals returned to piety.
Based on historical precedent across many cultures, the sudden disappearance of some visible attribute of the divine would rapidly be re-interpreted into some form that maintains social cohesion and cooperation. Likely, there would be a short period of theological fragmentation, but natural selection would rapidly select for the theological explanations that would keep society running.
Fictional Magic Explanation:
All of the above presumes that in the story world, the rings don't actually keep some form of chaos i.e. loss of predictability, separation of cause and effect, from affecting the human world. But if in your story they actually are some kind of supernatural defense, then human society would apparently face a very real upsurge unpredictable and non-casual events.
But likely, they would respond to the situation just like the real world historical societies did because prior to the rise of empiricism in the West staring in the 1500s, real world human societies believed that the world was largely chaotic and unpredictable. Only after Newton created the idea of the Clockmaker god, did the West gradually adopt the idea of nature as a type of predictable mechanism which followed rules that human's could puzzle out.
So, even if the disappearance of the rings did cause real chaos, the theological response would remain the same i.e. reworking the theology so that it continued to foster cooperation and functional social cohesion.