Microphone Not Working With FaceTime
I wonder if the iPhone is somehow detecting a pair of headphones in the device only when the FaceTime is active for video calls?

Steps to take:


Reboot iPhone
Try in phone app
Try again with FaceTime


Possibly also try a device backup and then restore either using your computer or iCloud for the backup.

Edit:

When reading your comment, you reminded me that there are 2 separate mics on the iPhone 5S. Microphones on iPhones. I believe the mic on the back is used for FaceTime and the front mic is used for phone calls as well as FaceTime Audio. Here is an iFixit guide list for the iPhone 5S.