Starter keeps turning and car will not crank.

many other things need checked like spark and fuel

Bad battery? If you are positive the battery is ok, maybe something is staying on and draining it. parasitic draw test YouTube

I always buy a service manual from advance or some parts supplier for every vehicle I own, that way the manual tells me how the parts are mounted to the motor and their location to the rest of the motor.
I am not sure where the starter is located on a Pontiac G6 but look above it if you have to use a flashlight, if your oil filter is near it, make sure it is tight and not loose.
Sometimes valve pan gaskets leak and run down the side of motor making us believe oil is coming from another place, just start looking and you will find it.

have a load test done on the battery as I suspect you have a dead cell or broken cell bridge.

Try a new bulb.

may be temp sensor giving false readings,,have a code check down for free at a parts store

I suggest that you verify fluid quantity and belt tension.

Is your charger working? Try jump starting it.

The remote start won't keep the car from starting.

If you have a check engine light on currently, that will not allow the remote start to function at all.