In hindsight, even if Robb Stark's will had never arrived, he'd probably still be declared king. Very few of the northern Lords wanted to be beholden to the south even before the last war. The only thing stopping them was the fact that they had fought with Robert Baratheon during the rebellion, and he had proven himself a formidable warrior. The stag king was also a brother in all but blood with Eddard Stark, so those thoughts did not surface while Robert and Ned were still alive.