I'd expect a noticeable delay with current local LLMs - especially visiting a site for the 1st time. But then they could potentially memoize their heuristics for certain designs, including recognzing when some "deeper thought" newly required by server-side redesigns.

But of course local GPU processing power, & optimizations for LLM-like tools, all adancing rapidly. And these local agents could potentially even outsource tough decisions to heavierweight remote services. Essentially, they'd maintain/reauthor your "custom extension", themselves using other models, as necessary.

And forward-thinking sites might try to make that process easier, with special APIs/docs/recipe-interchanges for all users' agents to share their progress on popular needs.