ADCs aren't sustained damage dealers trying for a deathmatch, and most of their skills are geared towards being a high damage glass cannon rather than a durable fighter.
During the laning phase, a Caitlyn (from your example) will try to use long range harass with Q or AA while trying to avoid damage altogether. If the enemy ADC comes into range, its the support's job to snare/block/shield/heal for her, that's why it is a dual lane.
Once past laning, in team fights Cait doesn't try to jump in like a tank, but will hover around the periphery dispatching targets from outside their range (a task in which she excels with her high AA range).
How does armor Caitlyn fare in either of these scenarios? Armor Cait will find it harder to harass an enemy support in lane to make them back earlier (thus forcing enemy ADC to be more defensive) due to her lower damage. If she manages to poke the enemy ADC, that will hurt less, so they can lifesteal it back faster. Lastly, she will have a harder time last-hitting mobs and may miss more CS, which is very important for an ADC.
Late game, that chain mail doesn't build into anything useful for her team fighting role. It doesn't give her tankiness to dive into fights, she will last maybe 1s longer while getting stun locked and unable to contribute. If she successfully stays outside enemy range, her actions will negate any benefit of having armor.
The quicker the ADC kills someone the less harm they do to your side ... so having an ADC who takes longer to finish off enemies is highly counterproductive to their main role.
As you can see, building damage is mandatory for an ADC to perform their role - they just cannot afford to ignore it in favor of other 'nice to haves' like armor. Hence why most ADCs need to go for risky builds and can only afford defense as a 5th or 6th item, since keeping up with the other ADC's damage is really important.