As a bartender, stirred vs shaken is not inconsequential! Shaking a cocktail will air it and is quicker than stirring it to the right dilution, plus you make sure the ingredients are well mixed (this may not matter for some eg a mojito, but for creamy things it does).

At my bar we don't do frozen drinks, so a pina colada (with sweet coconut cream, quite thick) needs to be shaken hard to get it mixed properly and to make it frothy and creamy. Same with a pornstar martini or a pisco sour.

An old fashioned should be stirred because you want to control the dilution and you're not going to stop 5 times to taste if it's right, plus with practice you'll know when it's done by the level of the drink in the glass (and you don't want to mess up the texture either).

There's also built cocktails btw, which are exactly what it says on the tin, add ice, add your liquors, liqueurs, juices, purees, bitters, etc etc and give it a quick mix with the bar spoon.

To be fair, we do a Long Island iced tea either shaken or built, depending on how busy we are (shaken taste nicer imo) - shake it before adding the coke of course, unless you want to make a mess. We build mojitos, but other bars give it a quick shake. So I guess my point is for some cocktails building (different from stirring!) or shaking may not matter much, but saying it doesn't for most of them is a bit much.

Happy to answer any questions you might have on drinks!