Reading this post pointed me towards a solution to my connectivity problem. I am posting my experience so that it might help someone else out.
Car: 2011 Hyundai Elantra Touring GLS Sport (Canadian market car, roughly comparable to USA GLS SE).
The Sport edition comes with the dual-DIN satellite-capable radio that also has iPod control capabilities. I believe this radio was used more extensively in USA cars.
The car has a connector block in the centre console storage compartment consisting of a standard USB socket along side a 3.5mm audio cable socket.
This car was traded in at a Kia dealership and I was in the process of purchasing it for my daughter. Having done research and knowing that the car stereo of this particular model was advertised as having an iPod interface, I wanted to verify this function as part of my pre-purchase checks. What I was most interested in was the ability to control the iPhone music playing functions using radio buttons and the steering wheel audio controls.
One end of the original Hyundai proprietary iPod connector cable has plastic block that places a UBS plug and a 3.5mm audio plug side by side (to connect to the combined sockets located in the centre console storage compartment), and at the other end it has the older Apple 30-pin iPod/iPhone connector.
The original cable that came with the Hyundai was missing. The salesman at the Kia shop knew that the same system was also used on some Kia vehicles of similar vintage, so he gave me the Kia version of the proprietary connector cable to try out.
First I tried using the Kia proprietary cable to connect an iPhone 4s running iOS 7.1. Everything worked correctly - the head unit sees the iPhone 4s as an iPod, music plays through car speakers, song names are displayed on radio screen, can control basic playback functions using radio controls and also steering wheel controls.
Then I tried connecting an iPhome 5 running iOS 7.0. The iPhone 5 uses the new 8-pin "Lightning" connector, so in order to connect this device I purchased a genuine Apple 30-pin to 8-pin adapter at Best Buy. I connected the adapter to the proprietary Kia cable and connected the iPhone 5 to the adapter. Again, everything worked correctly - the head unit sees the iPhone 5 as an iPod, music plays through car speakers, song names displayed on radio screen, can control basic playback functions using radio controls and also steering wheel controls.
Out of curiosity, I tested the iPhone 5 using a standard USB to Lightning charge/sync cable plugged into the car's USB socket, with no audio cable connected. Without the audio cable I was aware that I would not have any sound, but I wanted to see if I would still see the song names on the radio display. This failed. It tried to connect for several minutes, flashing the message "reading USB" on the radio screen and "Do you trust this computer?" on the iPhone screen. Eventually it gives up and the radio starts playing whatever radio preset it was on. (NOTE - I am not 100% sure that the cable I used for this test was a genuine Apple cable, so that may have been the cause of the failure.)
I did not have a standalone audio cable to test out UBS to Lightning cable + standard audio cable scenario, so cannot conclude definitively whether the proprietary connector cable (in this case a Kia proprietary cable used in a Hyundai) is absolutely required to make it work.
Anyway, I found a way to make it work, so happy ending here so far, although after reading this thread I am concerned that future iOS updates will mess it up.