No need to rebuild from ground up, sites just need a low data bandwidth fall back, perhaps with a new html tag indicating user is needing low bandwidth - much like sites used to provide a mobile ready area that was the same page load but simplified somewhat.
For the bleakest of disasters, bandwidth would be a premium but a lot can still be done without bandwidth hogging scrip overheads, so site developers just need to include a low bandwidth fall back for the basics - I dare say it might even be lest costly for the LLM scrapers if it were widly adopted. I'd suggest for an idea of basics touring some sites where each page has a small footprint [1] [2] [3] [4] ... I recall days on dialup 3 KB/s - a meg was a wait-a-while.
However I don't hold much hope, generally, things only happen out of necessity.