If in the restore you did, the backup of the device was restored as well, that could be the problem. Try restoring and setting up the iPhone as new to see what happens.
iTunes: Restoring iOS software
That's one thing I didn't try yet, and it sure goes deeper than what I already tried. Good suggestion, Sterling. I'll do that asap and I'll let you and the others viewers know the result.
Question: this suggests that something in my software is the cause (well, it couldn't be hardware since Facetime audio functions well). Any app or setting that might be responsible? Where to look?
Thanks anyway for your trouble in responding.
Ok, I followed Sterling's suggestion.
I did the restore to factory settings, and this time I did NOT restore my backup as well. So, the iPhone was like I had just bought it. Next, I did some test phone calls. But alas, the other parties still couldn't hear me.
(BTW: After the previous restore (that included restoring the backup) the other party could hear me for some 15 seconds, and then they stopped hearing me. Today it was dead right away.)
Thank you for the update!
It sounds like you may need service at this point but thank you for using the Apple Support Community. Due to the nature of your issue you may find more information by using another one of Apple's support resources - https://expresslane.apple.com/GetproductgroupList.action.