I think Harold marched North first simply because Harald and his army were first to arrive in England. And they were first because the same wind that carried their ships southward from Norway to England, stopped William from sailing northward from Normandy. Which leaves us with an intriguing question:
It was a near-run thing. The Saxons nearly won. If the wind had blown in the other direction, William’s delayed invasion would have landed before Harald’s and Harold’s undiminished army might have beaten William. But would he have beaten Harald as well?
And that quote…. is from the rulebook of my 1988 copy of the wonderful board game Britannia. :-)
I think it was a mistake on Harold’s part to march South in such a rush. If he would have stopped in London and given his army some time to recover, William would have had very little choice but to fight the Saxons anyway, because Hastings is not a good place to conquer England from when your enemy is based in London. It was different with Harald, who had a powerful local ally and could have taken the entire North if not confronted.