Let's suppose that some language decides to use the system package manager to install packages. They add support for RPM. Then for Apt. Then for Portage. Then for Pacman. Then for Zypper. If they're not too exhausted at this point and have any spare time between trying to keep things working on every flavor of Linux around, they decide they want people to be able to use their language on Macs. So they add support for Homebrew. Except that some Mac users don't like Homebrew and want to use MacPorts. Some others use Fink. So the intrepid language devs add support for those two as well. Now our poor maintainers are supporting eight system package managers that all have their own quirks and oddities that makes keeping all this working a massive nightmare. And all of this work is not done just once, it has to be repeated for every single package that someone creates for the language.
Oh wait, but users still can't even install language packages on Windows! You know — the most popular OS of all. Oops.
Perhaps this sheds light on why no programming language does it this way. Languages have to allow package installation on not only your personal favorite distro of Linux, but also on vastly more popular OSes — Windows and MacOS — that don't even have standard package installers that they can use. The only scalable way to do this is to allow installation of the language and its internal package manager on each platform and then use a consistent internal system that is abstracted from the platform for installing language packages. Which, not coincidentally, is exactly what every programming language does. Python is a little odd in that it doesn't have a single blessed package manager, and that causes a lot of problems, but using the system package manager is not the solution.