"but it will not install to anything on USB, it seems it has to be on the motherboard itself"
Are you sure? What error does Linux Mint give? Have you worked through the "try something else" options in the "install linux mint" program which let you specify drives and partitions. You should be able to install a non-live linux OS to an external HDD this way, and if the computer can boot from a live linux USB it ought to be able to boot from an installed linux USB external drive.
How about using another PC to install linux on an external drive, including put grub on that external drive in the very first partition, potentially with some UEFI partition too, then take that external drive to the damaged computer and have it boot from it.
In the end the external drive might be best to be an internal grade HDD or SSD, placed in a SATA-USB enclosure, this will have much betetr lifespan for write cycles than something sold as a USB memory stick or an external HDD storage device.