What you really want, to stream with OBS, is a second monitor. I have made do by playing games windowed and at lower than 1080p just so that I could see the edge of OBS with the critical info like the sound meter and whether it was working!
What might be nice is if VA could test OBS once per second and know if it was broadcasting or not, and then VA could speak the info to you when there was a start or stop without ever bringing OBS to the front.
That sounds to me like it might need someone who understands making VA plugins but I'm posting the idea here because some of the resident experts may be able to solve this in one line.
Another random thought - if you minimise OBS to the Taskbar and hover over it, does it change name/title to indicate online/offline because that should be something VA could test for easily...