I see. I had only ever single-clicked the Home button.
But now that I've tried it, I see the additional functionality. And yes, both the Channel Master and my trained OfA match the functionality of the Apple TV remote: Single click takes me to Home screen. Double-click pulls up an app switcher. Long press causes some sort of quick settings overlay menu to pop in from the right.
Only caveat is that it isn't flawless. The double-click on CM and OfA doesn't work if you click too quickly or too slowly; the Apple TV remote seems a little less finicky. But it does all work.