I'm having the same issue (Steam Deck screen goes black but TV doesn't get the picture). I didn't find a stable solution/workaround yet, so I have to unplug/plug from/to the Dock several times until TV gets the signal, and sometimes that doesn't help either, so I have to reboot Steam Deck (also can take up to several times) until TV gets the signal. That is quite inconvenient.

Okay.) I've solved the problem.
So, to reproduce if you have the same issue, follow these steps:
1) Switch to desktop mode
2) Open Screen Settings
3) Try to change the resolution to 1280x800
4) If the screen turns black and stays black, wait 15 seconds, the changes will be automatically cancelled.

This means that your monitor simply does not support this resolution (no idea why)

Steam does have a monitor resolution setting, but surprise surprise, it's only active when you connect a monitor. In my case, due to an update or my changes, the monitor resolution was 1280x800 for game mode and in this resolution the screen just stays black (which I found out with the steps below).

So, if you have the same problem and can reproduce it with the steps, you need to change the settings in game mode so that the external display is .
To do this, you need to remember the location of the "Settings" and "Display Settings" items in the Steam menu and do it blindly.
For me it was like this:
1) Connect the monitor
2) Press the Steam button
3) 2 clicks down => press A
4) 5 clicks down => press A
5) 4 clicks down => press A
This should turn on automatic resolution detection if you have it turned off and have set the wrong resolution.

You can also use Steam+RB to take screenshots with your monitor connected, and then disconnect it to check if you was in the right place and you have any custom resolution enabled for external monitor.

I hope this helps people solve similar problems.