Clearly that's not universally true, otherwise racing drivers would use automatic gearboxes. It may be true for everyday driving, but I'd need to see some citations for this claim.
Frankly, I'm a little dubious of automatic transmission. Perhaps I've just been driving the wrong cars, but I haven't found an automatic gearbox that can match a human being. For instance, if I'm descending a steep hill, I'll stick the car in a lower gear, but an automatic always chooses the higher gear.
Automatic transmissions avoid engine braking. If you want it, you need to choose it manually.
And personally I never do for the simple reason that brakes are a lot cheaper than transmissions. And I rarely descend a hill long enough and steep enough that brake heating is a serious problem. (And when I do I usually just slow down, and stay slow.)
I downshift on hills because that keeps more of my stopping power ready to use. Brakes heat up when you use them continuously, reducing their effectiveness.
Frankly, I'm a little dubious of automatic transmission. Perhaps I've just been driving the wrong cars, but I haven't found an automatic gearbox that can match a human being. For instance, if I'm descending a steep hill, I'll stick the car in a lower gear, but an automatic always chooses the higher gear.