The index would just point a local crawler towards hubs of resources, links, feeds, and specialized search engines. Then fresh information would come from the crawler itself. My thinking is that reputable sites don't appear every day, if you update your local index once every few months it is sufficient.
The index could host 1..10 or even 100M stubs, each one touching on a different topic, and concentrating the best entry points on the web for that topic. A local LLM can RAG-search it, and use an agent to crawl from there on. If you solve search this way, without Google, and you also have local code execution sandbox, and local model, you can cut the cord. Search was the missing ingredient.
You can still call regular search engines for discovery. You can build your personalized cache of search stubs using regular LLMs that have search integration, like ChatGPT and Gemini, you only need to do it once per topic.