Speed of reloading is another advantage the longbow has, but I think this article overstates it. While some crossbows do require using a stirrup or crank to load them, there are others that you can reload against your hips, and shoot from there, to increase your speed considerably, at some cost to accuracy. I know people who have managed to get 6 bullseyes at 20 yards on a crossbow in 30 seconds. Meanwhile, archers would not be firing at the maximum possible rate in battle; ammunition is a limited resource, and with the draw weights of warbows fatigue would set in quickly. Overall, with the archers they had and bows they had at the time, it is likely that the longbows were able to be a little faster than the crossbows, but it's not a night and day thing; and the range, accuracy, and penetrating power on the crossbows were better.
The simplicity became an advantage in a few battles, which came after substantial rainstorms that caused problems with crossbows more complicated mechanisms. But the main advantage was how cheap and fast to produce they were; you could easily arm a large populace quite quickly. In order to take advantage of the longbow, you had to do that; you needed a very large number of archers to effectively take advantage of longbows, while you needed fewer archers to be effective with crossbows. But because it was cheap and simple, it was feasible to do that.
I think that cost and simplicity of the longbows were their biggest advantage; speed perhaps a secondary factor, but the sheer numbers were likely to be more important.
There is, of course, an interesting parallel here with some trends in modern military spending. The Joint Strike Fighter is a technological marvel; one of the most advanced pieces of military equipment ever. However, they are staggeringly expensive, and not actually the best dogfighters in the sky. You wonder how much more effective spending that money on more and simpler weaponry might have been.