I might have solved my problem here; I waited a long time and went through several drivers to be sure.
Disable power saving on your monitor if you have this issue.
When I streamed Overwatch with power saving, the monitor turned off, and the stream would, too. However, when I streamed Paladins or Smite, the monitor never turned off, and I do not recall having Paladins or Smite stop streaming, so I disabled monitor power saving, and, then, Overwatch never stopped streaming...
This could have been resolved by updating drivers, but I saw no mention in any.