Does it hurt the brakes to apply them when going downhill?

My understanding is that that is more of an issue for those who downshift to force a decrease in speed rather than for those who set a lower gear early to prevent excessive speed.

Example near me is a small hill of about 400 feet with a road of about a mile – that’s roughly 7%.
If I down shift my automatic out of ‘overdrive’ at the start and accelerate gently at the start of going down, gravity will speed me up to just over the speed limit if I don’t tap a little bit on the brakes.
If I don’t downshift first, and even with gentle acceleration, gravity will have me pumping the brakes multiple times on the way down. Downshifting in the middle of the hill to slow down would be dumb.

And then there’s this: