I'm pretty sure Archery experience is earned based on damage you deal with a bow and not number of shots fired.
In general, you deal more damage when you fire more arrows and use better class of bows. So you are not wrong when you say doing more damage gives you more experience.
In my testing, there are only two things that affect experience gain, base damage of the bow and number of effective shots fired. Enchantments, potions and smithing improve damage, but do not contribute to experience gain. You don't have to believe me. You can test it yourself.