Battery activity

That will not stop streaming apps, which are actually running in foreground with the display off.

Background App Refresh manages apps that do periodic “housekeeping” by starting on their own. Apps that process notifications, or that provide streaming services, are not running in background; they are actually running in a non-display mode sort of between foreground and background. Actually, very few apps actually launch on a schedule in background or run continuously in background, so enabling or disabling Background App Refresh rarely makes any difference.

You can reduce Background Activity by going to Settings > General > Background App Refresh. Find the Roku app and flip the switch to off.

The Roku App will continue to run though in the background to provide the service when the screen is off unless you force quit it using the Application Switcher and swipe it up.

Most streaming apps will continue to run after you exit them, unless they have a Stop button. You can kill the app using the quick-launch screen it the app doesn’t have one.