One of the earliest reference to cocktails I can remember, I assume is like most people's earliest reference to cocktails.
Vodka Martini, shaken not stirred
While I did not necessarily understand what a cocktail was at the time, it sounded good.
Coming into the cocktail world, one could ask why are some drinks shaken and some drinks stirred? Does it even matter? Is it all just quirky history?
The answer, as all things, lies somewhere in the middle.
TL;DR
it makes the drink cold and it dilutes it slightly
Well there is a third thing, it adds air, which contributes to mouthfeel. But that is also very dependent on the drink's constituents.
That both method make the drink cold, I think is the most straightforward part of the equation and requires the least attention. It is mostly dependent on the duration of the mixing.
On the topic of dilution. While I would love to actually test this hypothesis, all I can offer right now is educated guesses and rules of thumb. Shaking, because of the friction involved, dilutes the drink more than stirring. The amount of dilution to which both methods differ, I cannot say at this time.
So then, why would you choose one method of mixing over the other? The rule of thumb is that cocktails containing mostly spirits, should be stirred. Cocktails containing, certainly, an egg white but also fruit juices, should be shaken. The cocktail alluded to in my first quote, contains mostly spirits and should therefore be stirred. I am of the opinion that it does not benefit from shaking, but then again, I am not a fan of Martinis.
And finally, aeration. While it is true that certain drinks benefit greatly from said aeration, the Ramos Gin Fizz being the epitome. To most cocktails, it remains a fringe benefit. The only time I see it as absolutely necessary is when mixing a drink that includes an egg white, as most sours do. The egg white and it's aeration are required for the foam floating on top of the drink.
In the end, the mode of mixing matters not so much for most drinks, in my opinion. You can make a perfectly good, shaken martini or stirred aviation cocktail. All that matters is that you enjoy it.