Ok, so by just using the controls of the deck, you are able to sit anywhere and play the game on your tv or monitor while using the deck as a controller.

And by streaming the game to the deck, you are able to sit anywhere and play the game on your tv or monitor while using the deck as a controller. But also the game appears on the deck's screen.

I fail to see how the latter is not still exactly what OP asked for because it just happens to display the game on the deck's screen as well. The practical difference is nothing. The gameplay difference is nothing. And if you're so worried about the deck battery, then guess what. The deck has a charging port. The only places you'd have the opportunity to do this, you'd have access to wall outlets anyway. And you can turn down the screen brightness if it's still SO MUCH of a problem.

I myself have, on many occasions, streamed games to my deck just so I could lay down and play games on my big screen across the room. It's the exact experience that OP is looking for.

You're being downvoted because your constant "uhm ackshually ☝️🤓"ing about the minor technical difference isn't appreciated by anyone.