Are we talking searching the URLs and titles? Or the full body of the page? The latter would require tracking a fuckton of data, including a whole lot of potentially sensitive data.
All of these LLMs already have the ability to go fetch content themselves, I'd imagine they'd just skim your URLs then do it's own token-efficient fetching. When I use research mode with Claude it crawls over 600 web pages sometimes so imagine they've figured out a way to skim down a lot of the actual content on pages for token context.
I made my own browser extension for that, uses readability and custom extractors to save content, but also summarizes the content before saving. Has a blacklist of sites not to record. Then I made it accessible via MCP as a tool, or I can use it to summarize activity in the last 2 weeks and have it at hand with LLMs.