My plan was to create the img of the disk, and use mkisofs to create a bootable linux CD from it. However I'm stuck between 'how to turn installation into file you can make an iso' part.

Create a bootable USB stick from Linux I was keeping the best for last. If you are running Linux, you can use the dd command (as in the instructions for macOS above) to create the bootable drive.

Thanks to open-source gem UNetbootin, you can quickly and easily transform a flash drive into the ultimate rescue tool.

Perhaps they referenced this guide to create a bootable Live USB on Windows, allowing them to test drive something like Ubuntu without making any changes to their PC.

We usually use Unetbootin to create a bootable thumb drive from a Linux ISO, but it turns out you don't even need a thumb drive if you want to boot the live CD on the same computer that has ...

YUMI Multiboot USB Creator allows you to create a Multi-Boot USB Flash Drive easily. Create Multiboot USB Flash Drive containing multiple operating systems, antivirus utilities, disc cloning ...

Will ImgBurn create bootable ISO/IMG files from floppies like the OP requested? In any case, I believe either WinImage (nag/no time-bomb) or Ultraiso (time-bomb) can perform the task.