You are absolutely correct Jaros, a good thought
The AOA is not important here (well, only if you are building of course), but the fact that a flat plate has a max lift coefficient just below 0,6 whereas a "curved" plate, or a cambered thicker profile typically has max lift around 1,0 to 1,3.
Because of the low AR a foil has, the high lifting foil will "lose" lift (tip vortices and loss at the tips), so the lift is not fully twice as high, but definitely way higher than a flat plate
If it was a glider wing (high aspect), the curved plate or profile would have slightly more than twice the lift of the flat plate
So yes, if using a flat plate (which will work fine), you need more surface area to get the same lift, although not quite twice the size.