There are two main reasons why you don't build armor as an ADC.
1st: The enemy Support as a target
Often you don't want to focus the ADC but also the enemy supporter (Let's just assume that it isn't a Leona or Braum but a Sona or Janna instead). Remember that about 70% of the Botlane work is done by the supporter. If you manage to kill the supporter (which is a lot easier if you already have bonus AD) the enemy ADC usually won't be able to do anything by himself vs 2. And like this you can snowball really hard, which is basically the thing to do as ADC.
Now of course you don't always want to focus the supporter and if the supporter is something rather tanky then don't even think about focussing him. So another thing you should take into consideration:
2nd: Your role in Teamfights
You are the Attack Damage Carry. The last word is the most important one. You bring the highest damage to the teamfight. If you waste your money on armor early on just to gain a small advantage for a small amount of time during the game, you won't be able to do your job right. Your jobs are Kiting, Positioning and Killing Whatever gets in range. If you build armor you just won't be a thread to anyone since ADCs heavily scale with AD and not with armor.
3rd: Farming
Well this isn't as severe as the other two points but... Farming is also a lot easier with more AD early on. You are able to lasthit better under the turret and with more Attackspeed you won't miss creeps that easily.