Should I downshift my automatic transmission?

You’re not putting any extra strain on the engine, but you might be putting extra wear on the transmission.

You’re trying to save inexpensive brakes by using a very expensive transmission. I don’t see the logic.

I generally get 50K miles out of my brakes, and I never downshift (manual or automatic) as a means to slow down.

Brakes are for slowing. Drive train is for going.

I agree with downshifting on long downhill sections, to avoid too much speed, but I can’t see doing it every day under normal driving conditions.

It is, however, your car, so you can drive it as you see fit.