I understand you would like to have the option to run Fedora Media Writer on a Linux system without installing it, in order to keep the existing OS clean.
While this option is apparently not available, as you have noticed, please note that Fedora’s Media Writer is the project’s way of easing the creation of a bootable media with one of the available Fedora editions/spins. The Flatpak packaging of the tool also means the host OS remains pretty much clean and secured (maybe more so than running a downloaded executable).
There are, however, other third-party tools out there which are used to create bootable USBs, such as balenaEtcher, which can be run without installation.
You can also have a look at this Fedora HowTo, which outlines several methods of creating an installation image.