The short answer is that compared to a long bow/composite bow a Crossbow sucks. The longbow out ranges it and has a vastly higher rate of fire, especially vs a heavy crossbow that requires a mechanical assist for cranking it.
The only real advantage to a crossbow is that you can train an inexperienced person to use a crossbow in a relatively short period of time (weeks-months). Where as a it takes about a decade's worth of practice to get someone to the competency level of your classic English longbowman. Since your typical character who is going to focus on a single weapon is going to more resemble the 10 years of training (since 10 yrs old) than the inexperienced millitia member. That's the way to go.
Also I suspect you can blame Robin Hood. Aside from William Tell (an opera, not exactly mass market), there aren't really any "crossbow" oriented heroes and he's really only famous for shooting an apple of his son's head. With Robin Hood as one of the all time classic adventure heroes (and a primary source for things like D&D) guess who things are going to be biased towards.