I hate the idea of paper statements, because it is just such a waste of resources. But, the reason above is exactly why I have all paper statements on all relevant accounts. Companies and/or the cloud are always more than willing to instantly cut you off for stuff you might desperately need. And I hope they aren’t charging you some $35/page to give it back to you.
That said, I now see a business idea. One that automatically downloads all statements from all accounts and digitally stores them for 70+ years (securely stored, of course).