CJN2
The Power button and the Volume button/Mute button on the Roku remote are "TV controls". They have no effect on the Roku device. The Roku devices really never turn off, they enter a sleep mode. The Home button is used to "wake up" the Roku devices.
With your Streaming Sticks, likely your USB ports power down when you are turning off your TV using the power button. Depending on how long it takes for your TV to boot up, you may be seeing the Roku logo on the Streaming Sticks when they are booting up. (if you connected the Streaming Sticks directly to wall power, you would not see the Roku boot up logo. You would just be at the Home Screen.)
Keep in mind that if you do not press the Home button when you are finished streaming, and simply press the Power button, the device will continue to stream away in the background using data. (until screensaver or bandwidth saver kicks in if enabled in Settings. Or if the new Power Savings feature/20-min inactivity feature is enabled under the Power Settings in newer Roku models.)
If you look closer at the behavior of your Streambar, you should be seeing the same thing.
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Normal/expected behavior for your Roku devices.
1. Use the Power button to power on/off the TV.
2. Use the Home button to wake up Roku device. Also remember to use the Home button to return to Home Screen when finished streaming.
3. Enabled Bandwidth Saver in Settings/Network/Bandwidth Saver/[On], to automatically stop streaming after 4hrs if no user input/activity. (ie. fall asleep, forget to return to Home Screen, etc).
4. Disable Power Savings Feature in Settings/System/Power/Power Savings/Automatic Power Savings/20-min inactivity unless you need this enabled for some reason. Causes some unwanted behaviors on various Roku devices.
5. If you enable HDMI CEC on your TV, and enable Roku's CEC 1-touch Play feature in Settings/System/Control other Devices, then pressing the Home button will simultaneously turn on your TV, switch/activated the appropriate HDMI input, and wake up the Roku device. Again, need to press the Power button on the Roku remote to turn off TV. Also remember to press the Home button to stop streaming before turning off TV.