Streaming From Linux PC to Deck While Docked

(I am recreating this post from Reddit here as well in case anyone has any insight or if I solve the issue, so I can add my solution to the non-Reddit body of troubleshooting knowledge)

I am quite a happy Deck owner and have had no problems that I couldn't tackle with a bit of Linux knowhow and willingness to do a google search.

But... I am trying something new. I have the official dock and have used it for months to play games directly off the Deck just fine. I have also used the deck to stream games off my PC running Linux Mint (Cyberpunk worked nicely). What I am trying now is to stream a game from the Mint PC to the Deck while the Deck is docked to the official dock.

I dock the Deck (wired to the network), and it can see the PC (also wired). I can click to stream the game as normal and it even starts the game on the remote PC. But that is where it fails. It sits on the loading screen for a moment, and then aborts back to the Deck library. I have the option to "connect" to the now running game, but it only pulls up the generic loading screen again for a few moments before dumping back to the game page in my library.

Games I have tried: Cyberpunk, Outer Wilds, Fallout 4. All of these run just fine when streaming direct to the Deck, but fail when trying to do it docked to a TV.

Any ideas?