GrapheneOS primarily exists to greatly improve privacy and security compared to the Android Open Source Project (AOSP). AOSP provides far better privacy and security than a traditional desktop Linux distribution. It has a strong mandatory app sandbox, an increasingly good permission model moving more and more towards case-by-case consent, broad use of memory safe languages throughout the OS and app ecosystem, strong MAC/MLS policies developed as part of the whole OS, modern exploit protections, verified boot with downgrade protection for the whole OS and far more. GrapheneOS starts from the already good privacy and security of AOSP and greatly improves upon it. We greatly improve the permission model, exploit protections and much more but we depend on starting from a foundation that's already decent.
Moving to a far less private and secure desktop software stack is going in the opposite direction from GrapheneOS. GrapheneOS doesn't exist to simply provide an alternative to mainstream operating systems but rather to offer much better privacy and security. We wouldn't be doing that if we were forking a desktop Linux environment and doing similar work for it. It would be nowhere close to the privacy and security of simply using an iPhone. That's a major part of why GrapheneOS is based on AOSP rather than it solely being about compatibility.
Desktop distributions are incredibly far behind on privacy/security and lack any clear path to achieving the same things. Every year, Android makes backwards incompatible privacy and security improvements as part of a new target SDK version. Android retains compatibility with legacy apps, but apps distributed through the Play Store (and other app stores to an extent) are required to move to the new target API level within around a year. This results in apps being forced to conform to a gradually improving privacy and security model. There's no such thing for desktop Linux apps but rather apps choose how much they want to participate in nascent sandboxing efforts.
GrapheneOS has near perfect compatibility apps from the Play Store via our sandboxed Google Play compatibility layer with the exception of banking and government apps. 90% of banking apps currently work on GrapheneOS because it greatly succeeds all of their security requirements and is only wrongly banned by a subset of those apps. These apps are gradually adding more anti-tampering and attestation checks for the hardware and OS, so maintaining compatibility has required us to gradually add more functionality working around it. We've also had to actively convince apps to stop banning non-Google-certified operating systems or to permit GrapheneOS and other secure options alongside doing it. A growing number of apps are choosing to stop banning using GrapheneOS due to pressure from our expanding userbase.
Waydroid uses namespaces and a compatibility layer to run an outdated fork of LineageOS. It's far from providing full functionality and compatibility with apps and nowhere close to the compatibility provided by GrapheneOS. More importantly from our perspective, Waydroid disables most of Android's standard privacy and security model through not supporting SELinux and other core parts of the security model. Android doesn't simply use SELinux as an additional layer of security with targeted policies but rather it's deeply integrated into the OS. It uses very strong whole system policies to implement the app sandbox, drastically reduced kernel attack surface and a lot more. Namespaces for the overall userspace environment are not a replacement for the app sandbox, kernel protections, verified boot and the rest of the security model.