Byanbcorny, fundamentally you have to use a Bluetooth headset, or some kind of Bluetooth device in order for everything to work as intended. Connecting a hands-free device not only means that you can hear the caller in the headset or speaker of your BT device, but more importantly, the microphone is "routed" to such hands-free device and this means the other party can hear you well.
Without a BT hands-free device you are using the microphone in the handset (phone), which is mostly not ideally placed while driving, nor is is designed by Samsung (or any OEM) to remove background noise and echo. All good hands-free devices are relying on some sort of "DSP" (Digital Signal Processor) for removing the background road noise and also minimize echo, that is why the phone itself is only good to use as a speaker phone during low speeds, when holding it near your mouth and/or in very quiet cars.
We will arrange some additional testing this week with the scenario you are describing where you have AUX-in on the car stereo but you want to use the S III handset speaker for phone calls. Since we have always believed everyone (?) is using a Bluetooth device/system for hands-free we need 1-2 days to check this out and revert back... Thanks!