It should work seamlessly since Steam Input operates at a lower level.
The app will receive the already-mapped input, so no additional handling needed.
The dreamcast port is based on the mentioned Re3 (it’s a fork), which can becompiled on both PC and PS2.
The author showed the gameplay on the original hardware some time ago. So I guess he adapted the controls as well. Sorry if I confused you.
On PS2 - yes because Re3 could be compiled for it. On PC it will work with modern controllers. It’s just a reverse-engineered port with some enhancements.
But on Dreamcast - it depends on the port.
They usually cut costs in unnoticeable for regular consumer way.
Like include some chemistry that easy to produce and makes it tasty, but bad for health.