Basically, the AUR works by downloading source code and running a build script that builds the app specifically for an Arch system, right? So why isn’t there an equivalent that works on most or all distros? Almost every time, compiling a Linux app is just running the commands that the developer tells you to, and for the uncommon times there are distro-specific differences, they’re usually listed in the README. For many userspace apps, even building across different processor architectures requires little to no change in the commands required as long as there’s a platform specific compiler. Why isn’t there a cross-distro system that can download source code and run a build script while knowing what distro-specific commands to use based on the developer’s instructions? Heck, compiling an app on the system you plan on running it on can unlock a little more performance by taking advantage of compiler optimizations for that specific processor, and it doesn’t take that long on a reasonably modern computer anyway, so why isn’t there an incentive to do it more often?
The real answer to “why” is that GNU/Linux users are so used to compiling applications themselves from source that most would consider this an unnecessary convenience. I agree with you, for the record, but I’ve had other GNU/Linux users look at me with a puzzled expression when they learn I use a GUI file manager. Most users are set in their ways.