Shaking or stirring a cocktail doesn’t just make it colder, it dilutes it as well. Getting this right is very often the difference between a good and bad drink.
I demonstrated this once when a friend complained that it was so hard to get a good cocktail, especially getting good ingredients. All they had in their kitchen was some gin and some other slightly floral novelty liqueur. I just took a spoon and a glass and stirred them something based on the ratios for a martini and they said it was one of the best cocktails they have had!
Balancing chill and dilution, even in more complex drinks, is essential.
A good way to experiment is to make batch chilled cocktails in which case you need to explicitly measure out the water you’re adding.