Did Hazō or anyone else ever bring up how it seems as if there could be a direct connection between the incipient collapse of the Seal holding back the Dragons and the appearance of someone with a bloodline marking him as the Heir to the Sage of Six Paths? My understanding is that these two events appears to be totally unprecedented. Recorded history is naturally extremely limited due to the apocalyptic nature of the setting, but I think we can safely assume that the awesome power of the Rinnegan and the destructive potential of anything sealed away by the Sage would still prove sufficient to leave an unmistakable historical impression. The absence of such an impression means that world has seen two events that had never before occurred over the countless millennium since the Sage was last active take place within what must have been at most a few decades. Doesn't it make more sense for there to be a connection than for this to be just an incredible coincidence?
If we assume that there is a connection between these two events than it seems as if we have an answer to why the Sage didn't take steps to prepare for the eventual collapse of his Seals. His plan would have been for his spiritual heir either restore the failing seals or gather a new group of powerful followers (as Pain did with Akatsuki) to find a way of implementing a more permanent solution. This plans seems like it would provide far more of a reliable solution to the eventual collapse of the Seals than relying on the accurate transmission of knowledge by a few clans. The only thing needed for this plan to work is for the Sage to have some means of transferring his knowledge and unique power into the distant future. The failure states of such a plan are limited to some kind of error in the transfer process or the complete extermination of anyone who could serve as a potential heir. There are far more failure states in any plan that relies on people following his past instruction. Consider all the challenges that a clan would have to overcome to survive and maintain an intact culture for the thousands of years that would pass before they are needed. Even if we assume that there is some kind of Bloodline or Artifact mental influence that could keep the clans indefinitely focused on their intended purpose, there would still be the strong probability of them simply dying out due to the incredibly violent nature of the world.
For me as well, Pein was my frontrunner explanation for why the Great Seal started breaking, primarily his heckhuge bijuu-powered ritual at Nagi Island, but I've since soured on that idea since, to the best of our knowledge, the Great Seal showed problems before the Nagi Island ritual happened.
Your take is a new spin on the idea though: Instead of the Great Seal breaking because of an action caused by Pein, Pein's Rinnegan manifesting because of the Great Seal starting to destabilize. It neatly explains how Pein might get his hands on the Rinnegan, but leaves little explanation for why the Great Seal went wrong other than the passage of time, which feels... I dunno.
It's always been the greatest coincidence of this setting that the quest started during Pein's time, when the Rinnegan was alive and something truly huge was in the works. This happenstance created a setting where 'heir of the Sage' Pein and 'fed inspiration by eldritch beings' Hazou coexisted at the same time, despite either one being a century-defining existence at the very least. Of course, OOC we know perfectly well why that happened, all the fun characters we know and love are in this timeframe as well and it just makes sense to start Hazou off around then, but IC you would have every right to look at Pein and Hazou happening at the same time and find yourself incredibly confused.
(This is perfectly fine: every story gets one free contrivance and that's in the story's premise. It's no grand coincidence that Naruto just so happens to follow the one ninja who turns out to be Ninja Jesus, nor should we be skeptical at the odds that the little muggle-raised Harry Potter turns out to be exceptional even by wizard standards. That we fled from Yagura's Mist as opposed to anywhen else is part of the premise of our story, not a strain on our credulity.)
Anyways, the thing about Hazou is that he's also going above and beyond in his own way. The other frontrunner explanation for the Great Seal failing was Hazou's proliferation of Skytowers to the Pangolins, leading to wide-scale and long-term seal use on the Seventh Path for perhaps the first time in millennia. This totally separate from Pein's also-once-in-a-millennia Rinnegan hax. If the question is what once-in-a-millennium-event the Great Seal is related to, both Hazou's seal proflieration and Pein's Rinnegan sound like plausible options to me.
(Of course, in the former case the line of causality goes from Hazou's actions to the Great Seal failing, while in the other case it's the reverse with the Great Seal failing leading to Pein's rise, but in each case we're left with exactly two once-in-a-millenium events: either Hazou and Pein or Hazou and the Great Seal)
But yeah, good thought with the idea that the Great Seal's failure caused Pein's rise! We'd then be back at square 1 for why the Great Seal started failing in the first place, but it'd explain the other big mystery of how Pein happened.