I do think it came out of convenience, not just in a stick being a 1/2 cup, but there are also convenient markings on the side of each sticks wrapper, indicating tablespoons, 1/4 cup, and 1/3 cup.What interests me is why a 'stick' of butter in the USA.
I can imagine American home cooks, back when butter was sold by the pound or whatever it was, being annoyed at having to dig out and measure 3 tablespoons, for example. Now it’s just a matter of counting three marks from the end and…whack! There’s my 3 tablespoons.