In a nutshell: It seems logical, but it turns out to not be more fun for anyone.
On one hand, yes, it seems to make sense that my young would-be-Robin-Hood PC might practice much more with longbows (or perhaps just plain bows) more than swords. On the other hand, it tended to remove more fun to the game than it added, and it opened up class balance questions without an easy answer.
Specifically, the option to specialize must be balanced between Fighters who go that path and Fighters who do not, or Specialized Fighters becomes synonymous with Fighters Who Do Not Suck. Not only is this a hard design problem simply within the Fighter class, it becomes even harder once we consider other classes.
Well, speak for yourself. I'm perfectly fine with having a fighter who wields nothing but axes (let's say the greataxe with the occasional throwing axe thrown in for a ranged option) for the entirely of his adventuring career. I think that is a "fun" achetype for play because I don't necessarily care about weapon-swapping for my version of the fighter.
I do understand the concern of balance. However, a reasonable compromise emerged in the ToB for 3.5 that I recall: could the warblade not concentrate for a period to swap the effects of his feats that worked on a particular weapon for those of another weapon? In a similar vein, 4th edition had the arena fighter, who could pick two different weapons and apply certain bonuses that applied to one weapon type to the other. If we took that concept and expanded it, perhaps we could come up with something that worked. Perhaps "weapon specialization" could be changed with a short rest spent practicing with the new weapon.
Or, if we want to keep weapon specialization specifically separate, simply have a specialization fighter as a separate build. Or make it a specialty. To be honest I find a lot of the current specializations that are viable for fighters to be rather dry, only the guardian one appears the slightest bit engaging but it has some problems with stacking too many effects on "reactions" when the protector fighter already wants to use his reactions for other things, and you only get one reaction per round! Making specialties that focused on particular weapon groups could be interesting as long as other feats were just as powerful.