Screen backlight not working - Logic board working on other screen

Your process makes it clear the issue is within the display assembly and/or the display cable its self.

I would have just used a DVM to measure the backlight power on the logic board. But you process also works!

So you'll need to inspect the display cable contacts for damage it needs to look like this

If that looks OK, you will need to inspect the cable its self for damage and may need measure the continuity between to pins across the cable contacts once you've removed the cable fully from the display following this guide Sostituzione gruppo dello schermo MacBook Pro Retina 13" metà 2014 Once the display is off slide the black clutch cover to gain access to the display cable connection

The connector is just like the logic board, also look for corrosion damage across the soldered pins as well often I see the connector its self damaged. People over did the cleaning of the screen so cleaner often dripped down damaging the connector and shorting out the solder joints.

And lastly, here's the needed replacement cable: 13" MacBook Pro Hinge Left - eDP Cable, Apple P/N 661-02360

If the cable is OK then you have a deeper issue within the display assembly, I find its time to just replace the part MacBook Pro 13" Retina (Late 2013-Mid 2014) Display Assembly