My guess is that swords are the most common quest rewards in SAO (it is called
Sword Art Online
), so as the floors go up you will definitely get a good sword from quest/drop and don't need to train a new skill.
Continuing with quests, in First Day he chose to do the quest for the Anneal Blade. As a beta tester he definitely tried out many quests, but he still chose to do the quest for Anneal Blade, the reason being it could be used up to the 3rd floor.
He most likely used 1h swords in the beta as well, so in the death game SAO has become, entrusting your life to a familiar weapon is vital.