It was the fact that the controller direction was based on the character's perspective, not the camera perspective. In my mind "up" should move you away/up in screen space.
The only other game I recall playing with static cameras was Resident Evil, which I haven't played in 20 years. I can't remember how they did it. Hmmm....
[time passes]
And now I'm back from reading the wikipedia about Resident Evil. You followed their method! It even has a name, "tank controls". Thinking back, I think I had the same issue way back when I first started playing. I suppose it's a thing you get used to?
Now I want to experiment even more!