If so, how do we avoid breaking the web while keeping privacy needs in balance?
One thing that I feel would help: Massive decentralization. Self-hosting of content and regular synching of the hosted content on the server sides; or tunneling, think duckduckgo.
Self-hosting would make knowledge storage more redundant which protects against (also partial) network blackouts.
On the other hand this knowledge is then harder to control. Removing or redacting content would have to rely on the particular sites to "pull the updates" from the upstream. Copyright will also be problematic: each site would have to make copies of content with the (probably commercial) intent of serving it to consumers.
Still I can't help to think we need more decentralization and self-hosting.