Cannot create link /etc/mtab~
Perhaps there is a stale lock file?

As a result, some of the partitions were not mounted and the system was pretty much broken.

But, no need to panic, just erase all the lock files (be careful not to erase the mtab file itself !) :

$ rm /etc/mtab~*

Now test mounting your partitions to check that you don’t get this message anymore :

$ mount -a

If it is alright, reboot and it should be fine.