No recently released (many years) player model turns off - at best they go into a low power state, and Auto power savings' additional "power savings" is minimal at best, and when enabled causes all sorts of stability issues (especially with HDMI syncing/etc), so its recommended to be disabled.
If you are referring to the device activity light - try configuring a simple Roku screensaver (e.g. Roku Logo Rain) with a 1 minute start time, making sure to go to Home when done streaming.
