That entirely depends on the culture of the area. Such generalized examples are inaccurate at best, as it exports a single 'national' practice to other nations.
So let me take this wonderful opportunity for learning, though rather than a wall of text I'll break it into bullet point format. If any fellow experts in the educational field wish to pitch in by all means.
1-Were most individuals across national boundaries hunters?
2-Can you train those already using smaller bows (to hunt foxes, hare etc) to use a bow twice, if not more, as powerful.
3-Can societies train multiple specialty troops?
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-Most individuals aren't hunters at this point, not in the capacity you imagine. The hunters we envision often hunted what they were allowed to hunt, small game. For this, an extremely smaller bow would be required, 40-60lbs.
-This leads to the second point you make, "grooming". Archery, as you can attest yourself, isn't a martial art.
Unarmed or armed martial arts where finesse plays more of a role than brute strength. Pure muscle mass is required when using war bows, and we have to make the distinction from short bows which are more like their cousins the compound bows for mounted ranged units.
In the end, the raw strength isn't something you can build in a few months for the necessary requirements of operating a war bow.
-Which leads me to the last point, "national" tradition. It takes an entire nation to set up, either willingly or by chance, a specific method that indoctrinates a younger population into a specific fighting style. Greeks weren't known for their archery, they were known for their carpentry and foot troops. Rome wasn't known for its archers, since they were farmers and thus favored infantry troops. Celts, through raids and mobile combat were renowned horsemen. That isn't to say they didn't have other troops that could be levied from their home base, but we have to take into account limited manpower. Once you raise a contingent of "X" troops, you no longer have that manpower to go elsewhere. Many individuals in England were renown to practice archery, but that was because they grew up doing so.
Ultimately, no all nations should have access to war bows. They didn't have the cultural or traditional base to draw from. Or we could give the options for all nations to do so but be it extremely expensive for it takes time.
I'm more of a proponent of choice, so I side with the latter.