AI will certainly be a challenge for web sites. I think we must accept that the end of the 2010s internet is upon us. Add supported web sites, that offer the user as the product, will start to slowly disappear. That is not really a bad thing though. The drawback for us users is that services will start to cost money.
I read a comment by Bill Gates recently where he suggested that most people will interact with AI through personal assistants. This actually feels like a good point from him. A PA that actually works will add a lot, to a lot of people. Which goes to your point about how normal users will stop interacting with the web though an unfiltered browser and start using AI to access the web. Companies who sell actual good and services online should be safe as their income is not dependant on advertising.
I suspect normal users will end up paying for the search engines used by the AI. On some form of tiered approach. Apart from a few oddball users, like us here in the fedverse who will find other ways to make things work.