Sometimes in the morning, when I'm a bit foggy and I can't recall any AeroPress recipe details; or if I'm too lazy to search for cool AeroPress recipes, I'll do a quick and easy one like this. It's an involved version of the very early days go-to recipe, when I wasn't paying too much attention to the details.
Back then I stirred more casually, not really counting how many stirs I did - now I notice it does make a difference in the coffee strength. And I'm still experimenting more with steep time.
However I do find the strenght you press can affect the coffee taste quite a lot. When I press much slower, the coffee is clearer and has less sediment at the bottom of the cup. If you find your coffee has a bittern note (even if you grind it coarse), try to press it very gently, see if this makes a difference.