>It is already painful to browse web sometimes using the non-"standard" tools (that is, not a Chrome with Google account signed in, not an EU/US residential IP). What if tomorrow Cloudflare checks will require attested and signed browser binaries?

If it's painful, that's not because of cloudflare. You can get past turnstile challenges using tor browser in a VM. You still might be blocked because of policies set by site owners, but that can hardly be blamed on cloudflare.