No, definitely not. I'm completely incapable of logging into several different services that have Cloudflare's protection (including their own website) if I use Chrome on my iPad. If I try on mobile Safari on the same device (which has basically an empty history), it goes through just fine.
Something is broken.
Sounds very dystopian and DRM-y for me. You would have to enforce this at the OS or even hardware level (because otherwise bots will just lie). And probably mandatory by law.
Unfortunately I don't think we can force devices to differentiate between "user-originated" and automated traffic without changing internet fundamentals and locking it down.
At some point balance will tip where an internet with only authenticated traffic will be seen as more usable and preferred by the masses, then anonymity is over. New AI systems generating unfathomable amounts of human-like garbage to flood all UGC platforms might actually become the tipping point fairly soon.
If a Cloudflare-like service is a required part of internet infrastructure, then each ISP and hosting provider should offer their own, equivalent service. By law, if necessary, if the economics don't work out otherwise. Because having a single company be the arbiter and monitor of who may visit any website is, well, bad.