When you have A, B, C, D, and E, all different webmasters on different parts of the internet making a webring of A -> B -> C -> D -> E -> A, things get really messed up when C stops responding to emails.
As others have said: you need a ringmaster for this to work. When C stops responding to emails, you tell B to update their page and point to D, for example.
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We have so many more technologies today than we did in the 90s. I don't know why anyone would look back at freaking Webrings when we have... I dunno... Wikis?
Now the Wiki administrator (A), could recruit B, C, D, and E to be moderators on the same Wiki. When C stops responding to emails, everything keeps going just fine.
Etc. etc.
I don't think anyone who actually lived in the 90s with directory services, Ask Jeeves, and Webrings would ever think about bringing those services back. Like, the good stuff were IRC, AIM, IRQ, USENET ?