An overlay setup part of the stock configuration. I guess if you knew you were going to change many things you could remove what would only be replaced in the overlay from the initial squashfs image, but that's fairly advanced - either needing an expansion and recreation of the squashfs on another machine, or a rebuild of LEDE from sources.
I do wonder if what's being seen is perhaps a behavior where the jffs2 overlay ends up read only when it becomes full of past changes.
If the changes made are known, a factory reset should get the overlay writeable again and then the USB storage could be set up. It may also be possible to create a mountpoint on some partition which is itself a ramdisk, and so still writeable - but you'd have to re-do that on every boot.
root@Omega-24CF:/# fdisk -l
Disk /dev/mtdblock0: 0 MB, 131072 bytes
Disk /dev/mtdblock1: 1 MB, 1150464 bytes
Disk /dev/mtdblock2: 15 MB, 15429632 bytes
Disk /dev/mtdblock3: 7 MB, 7929856 bytes
Disk /dev/mtdblock4: 0 MB, 65536 bytes
Disk /dev/mtdblock5: 16 MB, 16580608 bytes
For me everything is a bit learning by doing now. So I think to put this info's in to the wiki could help several persons. But before i do so, please correct if i see something wrong or to simplified ;)