I had an accidental go-live today (thankfully noticed quickly and it was very brief). I googled a bit and came across this old thread, so apologies if ressurecting it is not the correct course of action.
This setting shared in your screenshot is enabled under Settings, General Output. When I click start streaming in OBS I am prompted "Are you sure you want to start the stream" as expected.
The issue arises can be recreated with a Stream Deck.
I have a multi action button set up that includes the following 4 steps System: Open OBS -> OBS Studio: Scene -> Soundboard: Play Audio -> OBS Studio: Stream Start. If I press the associated button on the stream deck. the stream starts without the prompt from OBS Studio.
Is the setting visible via the API or whatever the Stream Deck software connects to?