This is one of those cases where I was asking for something specific (OpenCV+Python) when I really should have asked about OCR in general and seen what knowledge people had on the subject.
As mentioned in the OP, I wanted something that would allow me to analyze a screenshot for certain words and values and then make decisions based on that. It didn't necessarily have to be with OpenCV or Python. That was just the only way I knew of that made such an idea possible.
I'm much more familiar with C# and .NET so I decided to look for some sort of OpenCV wrapper that would allow me to program in C#. I found some, but while researching the topic further, I discovered that Microsoft has a Computer Vision library built in to the UWP SDK. So I can just have a line near the top of my code saying:
And voila! I've got OCR in my C# code.
Of course, this also means that I need to build/deploy a UWP app, which I'm also not very familiar with, but at least it's using a language and environment I'm familiar with.
All that said, while experimenting with the Windows OCR, my immediate need for this whole project was accidentally fulfilled so I haven't been pursuing it at the moment. But if/when I pick it up again I'll probably start with UWP and if that doesn't work then I may look more into OpenCV again.