Earlier quoted context omitted.
> I don't see why a signed bootloader starting a signed attestation engine wouldn't be trusted by third party websites. Do you mean a kind of Linux where root cannot do anything he wants? Like Android?
Yes, a kind of Linux like Ubuntu or Fedora that already boots with secure boot enabled with full support of TPMs and similar technologies. The kind of Linux 99% of Linux users are running today. More secure variants like Android, leveraging SELinux and such, help with sandboxing but I don't think that SELinux is a struct requirement.
Even after you manage to turn it on, it only verifies the kernel and cannot do anything about malware hiding in /usr. There is no Linux distro AFIAK that has verification of the entire system like ChromeOS, MacOS, iOS, Android and Windows have.