So, I'm having a similar issue.
Phone: iPhone 7
Carrier: AT&T
Operating System: iOS 10.1
Vehicle: 2013 Hyundai Elantra GLS
Issue: This has happened on my previous phone, an iPhone 6S, as well. However, since upgrading to iOS 10 and using the iPhone 7, the issue has increased. I feel that I'm a good candidate to test this issue because I drive 55-miles each way to work. So, I'm usually in my car for an hour each time.
What happens is, I turn my car on and turn the bluetooth on my phone on. After a few moments (rarely immediately; sometimes 1-3 minutes), the bluetooth will connect on my car stereo (I see the bluetooth symbol and it will flash, indicating that it's connecting) and then, it will disappear and drop the connection. It's very rare that this connects the first time. Maybe once every 2-3 weeks. This will continue (connect and disconnect) until it finally connects and then syncs my contacts. Sometimes, I drive an entire hour and my phone will never connect in my car. So far, this has happened twice in the past 2-3 weeks.
That is the main problem
Another problem is syncing and downloading my contacts to the car stereo. This usually takes anywhere from 5-10 minutes. I have 42 contacts in my phone, and remove my call history quite often. Therefor, it should not take this long ever.
I've only ever experienced this issue since Hyundai updated my car stereo's software. A previous update would instantly connect without any issue whatsoever except that every 15-minutes, my phone would go from hands-free back to the phone. The person on the other end would not be able to hear me, but I would be able to hear them. So it too me awhile to learn what was happening. To resolve this, I'd go into my phone's audio setting during that call and switch it from the phone back to hands-free. And once I did that, an internal timer would go off and I would need to do this every 15-minutes. So, if I noticed my phone getting to the 14-minute mark, I'd do it quickly so I wouldn't have the problem. And then, I'd have to keep my eye out for the 29-minute mark as it always happened in 15-minute intervals. The current version corrected that annoying issue. However, now, I'm experiencing the connect/disconnect issue.
Steps for Resolution: I've unpaired, paired, deleted the phone from my car, deleted the car from my phone.... more times than I care to count. This has not solved the issue.
I've also read in this forum about the Airplane mode trick. I've tested that theory out today and it does not work. In Airplane mode, I'm still experiencing the connect/disconnect issue.
Another theory is to turn LTE off on my phone. I've done this and also tested that it does not work. It still connects and disconnects.
I've even tried both of those, pairing and unpairing several times, and it still does not work.
Tomorrow, I have an appointment with Hyundai to replace my car stereo with another one. It seems that I have this problem more than most people and it's very frustrating with a new phone and a fairly new car. I hope this fixes it. I'll check in with an update, however, I don't see this fixing it at all.
My wife has an iPhone 5S on iOS 10 on Verizon and I plan on testing this out with her phone tonight and see if it works. But, for now, this is an ongoing issue and problematic.