“internet indexer” should really be a government / EU / UN sponsored public utility thing. And then you have front ends that can improve the search query and filter and sort the results.
I don’t know how much it would cost / how large a data center it would take to cache the web pages like google does. But there is just no reason to have so much wealth / advertising drain from the public for such a fundamental thing.
Well they will probably be massively over capacity. But then you could run the for half a day during sunlight hours and just use solar. Solar PM and wind kite power basically gives us near infinite super cheap energy at certain times, so you could still make use of those compute centers. Turn them off during the night, modernize the cooling and noise dampening, add some solar and you can use the compute to run models to create new medicines like anti-cancer meds or protein folding. Unlikely that’s going to happen but that is what a sane civilization would do with all that compute.