I do still whip out my AeroPress from time to time if I'm making a really fruity coffee, because I feel like the AP method is better at extracting those flavors. But for my usual morning cup for myself and my significant other, French Press is king.
> To my brain, AeroPress == French Press minus the "fine sludge".
I haven't really noticed "the sludge" in my FP since I invested in a Barata Encore canonical burr grinder. It's able to grind the beans coarsely and evenly enough that you don't get the sludge (which is usually the unevenly-ground fines that slip past the mesh filter). Additionally, paper filters trap the coffee bean fines, but they also trap the oil, which is desirable in your coffee.
But I will concede that the AP is probably the most affordable option for a good cup of coffee. You can pair it with a cheap blade grinder and still get a decent cup out of it, while poorly-ground coffee in a FP can make for a "sludgy" cup.