As of December 16th, there's a beta Steam update that adds support for non-Steam game configs, and possibly moves stuff around on the drive (meaning that the Steam Controller Database client might not work).
What does this mean for the Steam Controller Database? Well, I'm not sure. The new Steam beta is a bit buggy, and it's hard to say what's working as intended and what's just broken/not working yet. I'm going to be monitoring things as they progress and will post updates as I figure things out. For now, I *think* you should be able to use the configs here, but you might run into issues where configs for non-Steam games just don't show up as that's one of the known bugs with the new Steam beta.