I've actually tried several of the things you mention, with no success. I deleted my phone and re-paired it several times and that didn't resolve the issue. I restored the factory settings in my car, that didn't work. Each time the phone successfully pairs and my contacts are uploaded, but when a call is placed, it automatically goes into PRIVATE MODE, thus deactivating the handsfree (bluetooth) capability. I also paired the phone in my husband's Tuscon (where it hadn't previously been paired). It was successfully paired, but the handsfree mode is still inoperable (same exact issue occurs). His phone (a Droid) works fine and does not have this issue. It works in his Tuscon, and in my Sonata as well. The issue is definitely with the iphone.
After trying all of these things, I then took my car to my local Hyundai dealership. There, I was told that they were aware of several others who had been experiencing issues with the iphone specifically....that apparently a rececent security release from Apple had made the iphone incompatible with Hyundai bluetooth navigation systems, and that they were currently working on a fix/upgrade but couldn't tell me when that would be available. And that they were sorry, but there was nothing else they could do.
I'm open to any and all advice.