Weird, for like the last 3 or 4 xpacs I played, I think a weapon was one of the first quest rewards they gave you and it may have even been a blue. Like the intro quest for the xpac.
I've done both ways in the past, take my primary or first alt and go around getting a few pieces of gear before leveling up, or just loading up the optimal leveling path add-on, which auto accepts quests, tells you where to go with arrows, and has prompts on what to do. Following the guide and not stopping to "gear" up was always faster. Hell, once I had a piece of gear in every slot from the new xpac quest rewards, I wouldn't even look at gear rewards until 3 levels from the cap. The mod had an option to auto pick gear by gold reward.
What I am saying is the gear hardly mattered when leveling up via quests, and the add-on that gave the optimal path was the fastest way to level up, if that is the goal (in the past 4 xpacs, at least).