It doesn't seem like an especially hard lift to get ChatGPT to start indexing websites.
That is a very bold claim unless you've seen their architecture. Maybe you have, or read something to that effect, but citation is needed.
* poor coverage for post-2021 data means it's not trivial to update the models regularly
* wikipedia: "ChatGPT ... can’t currently rank sites for reliability, quality or trustworthiness" - to be able to train on all kinds of web data requires a proper ranking system, which is basically rebuilding something like PageRank or an alternative from scratch (and fine-tuning it for decades to filter out garbage). According to this quote, they don't have that.
* incrementally updating the web-scale data on a daily-ish basis is a big undertaking, Google had to invent multiple systems from scratch just to do that (e.g., had to create Percolator based on BigTable and MapReduce, all three invented by them just to do indexing) - and that's just a ranked index, not a machine learning model
And if it is, what stops Google from just ... incorporating an LLM into their existing search offering instead of having their lunch eaten by this hypothetical IndexGPT? It's not like Google lacks expertise in LLMs.
You've also not addressed the more fundamental question of why an LLM would even be good for this.
You'd think that, but it's the classic innovator's dilemma. Google would have to modify their product to generate less revenue now to maintain market share. Except that market share will decline regardless as competitors rise, so they'd be cutting revenue just to slow the decline of market share. Alternately they could release a new AI product that cannibalizes their own search profits.
Either way, for a large multinational obsessed with quarterly profits and the stock price, it's very hard to overcome internal resistance to do either of those things.