I'm trying to install the KDE spin on a new ThinkPad X1 and I just get the grub prompt instead of the live OS

I used the “Disks” app on my old Fedora 35 laptop to write a fresh download of “Fedora-KDE-Live-x86_64-35-1.2.iso” to a thumb drive. The sha256sum verifies. be0de125c785e52721d858d7a0cf70b6f50c099553296c0efe6df361ab21f017
I boot the ThinkPad and press f12 when I see the logo, use the arrow keys to select the usb volume, hit enter, and I see the grub> prompt. What now?

I suggest you follow the readily available instructions to write the iso to the USB and ensure it is bootable. From fedora you can use dd, fedora media writer, k3b, and many others. From windows you can use rufus and others.

You need to properly burn the image to the usb or it will not be bootable. You also need to know whether the hardware is using uefi or mbr boot mode, and select the proper mode for the usb device from your boot menu. On my system which I boot in uefi mode the boot menu shows me the USB device twice. Once as usb and the second is usb with a uefi overlay so I select the one with the uefi overlay to boot and install in uefi mode.

Okay so you’re saying the problem is probably that I used Gnome Disks to restore the ISO, and if I use dd then it should work?

That is a possibility. I have never used gnome disks for that purpose.