As I stated above, a square with 4 in the center would not have sums of 15 but rather 12. So 9+2+1. But yes, as stated above also: It doesn´t work. I understood the puzzle the same way you did, but Lorisaur apparently tried an alternative - and I think it´s perfectly fine to explore other possibilities.
To make it short: You are right, it is not possible - especially if you follow the instructions closely (Lorisaur: " I paied [sic] too little attention "). BUT: It´s not "really that simple" as you stated.