> How do you unmount your OS’s disk while keeping the OS running to be able to overwrite itself?

I went down a similar rabbit-hole myself, with the goal of safely replacing the Linux installation on a disk that a machine is already running from (e.g. replace a VPS's setup image with one of your own) without needing a KVM-style remote access tool to the console.

The problem there is if you directly modify the disk when a filesystem is mounted on that disk then all bets are off in terms of corruption of the filesystem that's already on there and also the filesystem(s) you're writing over the top.

My solution was to kexec into a new kernel+initramfs which has a DHCP client and cURL in it - that effectively stops any filesystem access while the image is being written over the disk, then to just reboot.