Ah, I just found it, but it's on the website only under "playback settings." You cannot turn it off from the app clients, or at least not from the Apple TV one. I think this still shows an intentional act by Netflix to make this hard to do, because most people are not using Netflix from a web browser / computer.

You also cannot turn off the auto-playing previews with sound.

For me, turning off auto-play worked on both my Roku and Visio TVs. I actually turned it off on a per user basis.

As for the auto-playing previews, my partial solution is to hit the mute button when browsing the Netflix home screen. Still gets annoying when trying to read the text description of a show, but it disappears from the screen too fast and replaced by the preview.