I'm most familiar with 3e and you needed a specific bow based on your strength. No propelled ranged weapons added strength to damage: cross-bows, short bows and long bows. Throwing weapons added strength: daggers, axes etc... No matter what, you used Dex to target and strength for damage.
You could buy compound bows but they had a rating, +1 to +5, which was the maximum extra damage you could do based on your strength. So, if you strength was 16 and had a +2 compound bow, you could only add +2 damage(not +3). But if your Strength was only 13, the bow was unusable because the weight of the pull was too much for your skinny arms.
It made it tougher to play a ranged character but, in earlier editions, you could make up the damage with magic items (+5 damage). You still might be able to make up the damage because, in 5e, even if you change the damage stat to Strength, Damage in 5e is about stacking dice and not stacking modifiers(although, it still helps, modifiers don't get multiplied in a crit). And there's lots of ways to stack dice.