UPDATE: Solved. Maybe somebody will find it helpful.
I can proceed with installation on my NUC.
Surprisingly I had to switch to Legacy Boot mode instead of UEFI Boot mode and keeping Secure Boot disabled it allows installer to initiate.
You can also do it with gnome-disk-utility installed. Right click on the ISO, then in the popup menu you can choose Open With → Disk Image Writer.
This is how I do it from my Fedora 35 install anyway. I haven’t found Fedora Media Writer any newer than something that was released for Fedora 24 when I googled (although it seems it’s available to install with flatpak). I really liked the way it was possible in Linux Mint using the USB Image Writer.