These questions are constant. I do think you bring up relevant issues, but they aren't quite showstoppers.
Websites allow SE crawlers because (a) whatever traffic they get is better than not traffic (b) because allowing crawlers is default and doesn't cost anything and (c) google/bing don't negotiate. They are one, sites are many.
This has already played out in news. News outlets wanted Google to pay for content. Google (initially) responded by allowing them to opt out of Google. Over the years, they have negotiated a little bit. Courts, in some places, forced Google to negotiate... It's news and politicians care about news specifically. Overall though, there have not been meaningful moments where people got pissed off with Google and blocked crawlers. Not newspapers and not anyone else. Site owners being mad doesn't affect google or Bing.
What does matter to search engines is walled gardens. Facebook pioneered this, and this does matter to Google. There is, in a lot of cases, a lot less content to index and serve users. All those old forums, for example.
These are search problems, and GPT-based search will inherit them. ChatGPT will have the same problem recommending the best air fryer as normal search does. GPT is a different way of presenting information... it's not presenting different information.
RE: Lawsuits. Again, history. Youtube, for example, started off with rampant copyright infringement. But, legal systems were primitive. Lawyers and legislatures didn't know what to do. Claimants were extremely dispersed, and would have had to pioneer case law. Ultimately, copyright took >10 years to really apply online and by that point youtube and other social media was entrenched.
The law lags. In practice, early movers are free to operate flawlessly and they get to shut the door after them. Now that Google is firmly entrenched, copyright law serves as one of their trenches.