The way i understand armor is, that each 1 point in armor is increasing (effectively) hp by 1%. Making the armor a +40% hp.
It is true that
effective hp = (1 + armor / 100) * base hpand therefore adding 40 armor will increase effective hp by 40% of base hp, but not 40% of effective hp (unless effective hp = base hp, i.e. armor = 0). That is, a champion's ability to take damage is increased by less than 40% by buying a chainmail.
Caitlin has 22.88 (+3.5 per level) armor, so 41 at level 5. Her standard rune page contains 9 greater seal of armor, adding another 9 armor, so she has about 50 armor before buying that chainmail.
That is, her effective hp without chainmail is
150 / 100 * base hpand her effective hp with chainmail is
190 / 100 * base hpso effective hp has increased by a factor of 190 / 150 = 1.267, i.e. 26.7%.
Now, what would the benefit of the pickaxe be? Let's do the same calculation for her damage output:
damage output = AD * AS + skill damageskill damage for an adc also scales strongly with ad, so let's simplify this to
damage output = AD * (AS + c)Cailyn has 50.04 (+3 per level) AD, so 65 at level 5. Her standard runes include 9 greater marks of attack damage, for another 9, and she probably has a doran's blade as well, adding another 7, for a total of 81 AD.
By buying a pickaxe, she increases this to 106 AD, i.e. her damage output increases by a factor of 106 / 81 = 1.308, i.e. 31%.
Therefore, a Caitlyn that buys a pickaxe will do 31% more damage than her clone who bought a chainmail, while the latter can take 27% more damage instead.
In terms of surviving a fight, the items are therefore about equal. However, having higher dps makes it harder for a losing enemy to retreat to the safety of his tower, increasing the likelihood of kills. It makes last hitting easier, resulting in a gold advantage. And it creates synergies with any attack speed, critical hit, or lifesteal items Caitlyn buys after that.
Therefore, an ADC is correct in not buying armor.