Ok, this is a little ironic but it turns out we had root access to the tablet all along since post #2 by the OP. It only took me four months to realize that.
Below, in PART 1, is how NOT to root the tablet (if you choose not to do so) and, in PART 2, how to root it by hand (if you still want to root it).
PART 1: How to access everything on the tablet, including /system without rooting it.
As mentionned in the by the OP. There is an image of the firmware of a similar tablet that can be obtained on www: iviewus.com/firmware788TPC/ Get the file.
Burn the image onto an SD card with PhoenixCard using the startup
Boot the tablet from the SD card: Insert card, press (and keep pressed) the return button, turn the tablet ON, release the return button as soon as you see the A31 logo.
The A31 boots slightly differently from the other Allwinner chipsets. More info here: linux-sunxi.org/BROM if you are interested. You should see an "IVIEW" logo during the boot and you should be able to acess two of the partitions on the SD card once it is booted. If you see the NOBIS logo and get an error saying damaged SD card, you did not boot from the SD, try again and make sure you are not booting in "Quick Boot Mode".
You might not realize it yet, but you are essentially done. Congratulations!
What just happened is that you booted the boot partition on the SD card which, as it turns out, loads a rooted kernel and ramdisk to your RAM. The nice thing is that the rest of the system is mounted from your own NAND partitions. This is why everything looks the same. Nevertheless, you can have full access to your tablet using ADB.
If you want to modify /system on your NAND. Connect the tablet to your computer via USB and fire up adb. The following commands will enable writing to /system:
Note that the tablet is NOT rooted and that restarting it will bring you back to normal. You might decide that you do not need to root it any more at this stage. Otherwise you can use your full access to permanently root the tablet by hand.
PART 2: How root the tablet by hand
Install superSU from the Play Store with your tablet booted normally. It will complain about missing su binaries if you try to start it. We will fix that in the following.
On your computer, access the 805 MB system partition of the SD card (an ext4 filesystem, mounted from /dev/mmcblk0p7 on my Linux) and copy the su binary from the /xbin directory onto your computer (in the same directory where you have adb). Alternatively, get the su binary from any other source you trust.
Restart the tablet as in PART 1 and follow all the instructions to get rw on /system.
Run the following code in your shell (bash or CMD.exe):
adb push su /system/xbin/su
chown 0:0 /system/xbin/su
chmod 6755 /system/xbin/su
ln -s /system/xbin/su /system/bin/su
Done, you are rooted. After restart, you can let superSU update the su binary and you are all set.
PS. I am not allowed to post links, I am too new here. I hope it is not an issue.