Ollie, that is a great observation. I do remember Rickard Karstark challenging Robb to do the deed himself. I have two guesses as to why this did not happen. First, Eddard was so stunned by Joffrey's announcement that he could not think rationally. Second, Eddard's death happened in Arya's POV. She was not close enough to make out anything Eddard said after the crowd erupted upon Joffrey's announcement.
And if Eddard did ask, I think Ilyn Payne was under orders from Littlefinger to proceed without delay. I just made that connection... Payne was paid by Littlefinger. Jaime discovered this in AFFC while talking to Rennifer Longwaters. When Joffrey called for Eddard's head, Cersei, the High Septon, and Varys all immediately protested but Payne and the Hound ignored them and carried out the deed. Littlefinger was not on the podium, as far as I know. But I would not put it past Littlefinger to have persuaded Payne to ingratiate himself with Joffrey and to take Littlefinger's money, if by some chance Joffrey reneged upon the deal.