How to recover a System partition that won't mount
My not-so-great solution ended up like this:


ClockWorkMod, installed with Odin according to the instructions (Odin seems to be slightly un-official so you're unfortunately always downloading from someone's Box or google drive, which is a bit uncomfortable)
ADB shell, (on PC: install java SDK, install android development kit)

Mount the SD card with CWM
In the shell, using DD to clone the unmountable partitions I wanted onto the SD, for later recovery
using CWM to mount the SD as a USB drive, moving the piece-of-partition to the PC
Repeat

Download fresh firmware from Sammobile and flash with Odin
Scratch through the partitions with a hex editor, to find the data I needed. Fortunately this was just text, things like pictures would have been harder.


A few features were still not working, WiFi for one, so the phone had to go back to the shop to be flashed with official firmware. At least it had the official bootloader on it at that stage, not CWM.