The GrapheneOS instructions are really strange, only meant for Ubuntu and more. For example, bsdtar is not needed, just use unzip.
The WebUSB stuff only works through Chromium, I dont think this has changed.
As GrapheneOS has its own way, there is this tag.
I did an install recently and it worked well.
Instead of a rootful toolbox, download the official Google binaries to ~/.local/bin/, as the Fedora fastboot doesnt work for GrapheneOS / modern Android / whatever anyways.
Follow the “without installing to OS” guide for the correct instructions.
@vgaetera told me that the guide is likely still not perfectly correct, but it works.
You need the udev rules to theoretically make the Browser install (rootless) possible.
But it didnt work at all, not even on another phone, last time. Fedora Kinoite CLI was the only one working. This was some strange bug, I installed GrapheneOS already from another GrapheneOS phone.
You could use another Android phone with a Chromium-based browser.
To use the web install, after installing the udev rules, layer Chromium, but only if you want. The CLI install works fine.