This seems very impractical given the way the internet currently works. Most startups use dozens of SaaS products, let alone more basic/foundational things like global CDNs. You're being logged at every step of the process if only to prevent spam/DDoS/etc.
What you're asking for would require a fundamental restructuring of the internet, and of software business models, and a lot of other stuff. I can't see that happening any time soon.
In the meantime you can try using Tor, but good luck not getting blocked on half the websites you want to visit - and you can't blame the website for that (they need DDoS/spam defence).
Not only the internet, this is impractical given how any business works. Even a brick and mortar store is sharing aggregate customer buying habits with its supplier based upon products it buys from them.
When I visit a website of some business, I provide them with an IP address for use during the session (because of the way TCP/IP works). I'm okay with said site using some kind of load-balancer, DDoS protection or what not, as long as the business takes full responsibility to keep my personal information private unless I specifically indicate otherwise (opt-in[1]), for example using a form on the landing page. I believe that this is the true intent of the GDPR in this matter.
What you're asking for would require a fundamental restructuring of the internet, and of software business models, and a lot of other stuff. I can't see that happening any time soon.
In the meantime you can try using Tor, but good luck not getting blocked on half the websites you want to visit - and you can't blame the website for that (they need DDoS/spam defence).