We have Roku and are able to use it to fall asleep without the streaming continuing after the TV goes to sleep. The "secret" in our case includes not using the Pluto channel -- which, yes, does continue streaming as it's live TV -- but watching shows that part of the "Buzzer" network (more about shows in a minute).
We discovered Buzzer when we had Cox cable and there was a Buzzer channel, and the main shows on it were game shows from the 50's through 70's (Tell the Truth, I've Got a Secret, What's My Line, and Gene Rayburn doing Match Game). As the late night shows pivoted to highly-political we lost interest in falling asleep to that, and started watching Buzzer. Great way to fall asleep.
When we cut the cable cord we saw that Amazon Prime had the same shows, and that when we clicked on a show it began with the Buzzer TV logo and announcement. I can confirm from our experience that when we turn on the TV the next night the Roku is back at the main screen, and the number of shows "streamed" essentially matches the amount of the "sleep" timer.
We recently put a much smarter TV in the bedroom (which has an app for Amazon Prime) and the same thing happens, even without the Roku.
YMMV, but hopefully this helps.