Most of the jailbreak apps are acknowledged to have better screenrecording features. Once my brother bought an app off the appstore and the ‘screenrecording’ consisted of taking loads and loads of screenshots and stitching it together to create a video. Appstore apps generally aren’t allowed to properly record the screen.
The link shen_w00 gives you lets you install apps for jailbroken devices without being jailbroken but you can only have one app of the kind using the code signing hack because Apple cracked down on it and revoked the certificate. Which is probably why it doesn’t work on iOS7 and the fact you need to change the date to make it work.
So I conclude that since it’s impossible for you to downgrade, your best bet is to mirror the screen with something like airserver or reflector and then record it.