Can Apple Watch 3 with cellular work even if iPhone is turned off

SMS message do need the iPhone to be powered on and accessible via cellular, WiFi or Bluetooth. It is a difference in the radio in the Apple Watch and the iPhone. SMS text messages requires the iPhone radio.

Suggest you read the following

https://support.apple.com/en-us/HT205547

and pay close attention to the footnotes

Yes, if you have a cellular plan with your phone carrier.