I IAC (Idle Air Control) motor would only effect the speed of the idle. Usually sputtering is caused by miss fires, either random or sequential. If I understand what is going on, when the engine is left at slow speed for a while, it starts sputtering. If the engine is then rev'ed up it cleans up. There are 2 things that come to my mind; Dirty plugs, or dirty/leaky injectors.

An oil or carbon fowled plug will misfire when it is cold. If you rev it, it heats the plug up and puts a thermal load on it, temporarily cleaning it up.

A dirty/leaky injector will behave the same way. If it has a slow leak, it will leak all the fuel in the rail out; flooding the motor. When started, there may not be enough air flow to clear the flood. By reving it, the airflow is increased and the flood cleared. With the cylinder now running warm, it will run rich, but not misfire. A dirty injector is on the opposite side of the spectrum, but it's symptoms can be the same.

Considering how easy it is to remove plugs compared to injectors, I'd start there. Pull the plugs and inspect them. If you haven't replaced them in a while, spend the big $20 and throw some OEM plugs in it.