Ok, so Harolds shield wall holds all day long. The Normans leave the field of battle. What does Harold win? Because the Normans decide when they fight and how long, I do not think it would have mattered if they lost that day or not.

Because the Normans have mobility, they can decided when to attack or to even attack at all. The Saxons on the other hand, would have won a tactical victory. That is why I always wondered if the breaking of the wall by surges of unruly warriors from the shield wall was not really an attempt by Harold to counter-attack that just failed horriblly.

I just can't see a Saxon victory making a difference. At Stamford Bridge, Harold was on the attack and that is why that victory ment something. At Hastings, William was on the attack and only a Norman victory would have had any chance of a stratigical difference.