There was some overlap between the webring age and the early search age, but once search became entrenched and useful, webrings faded. Blogrolls survived for a little bit longer, but it was search.
Specifically, once search became the way you found the first page/site to begin with. Before search as default, you found sites in a bunch of scattershot ways: advertisements, word of mouth, lists in books, or lists on websites that updated periodically (that you had to have found/heard about one of the other ways) for example. Then you crawled out from there because that was the only way to find things. You had to either know the URL or use a link. And not all of the links/sites in the webrings were good.
Once search got good enough, people found the initial site via search and instead of taking their time clicking through a webring which might at any point lead them somewhere dead or useless, it was quicker to go back to the search page to find something else.
Page access went from being a chained together web of back and forth links to a 2 step process of search -> page.