All the same, that doesn't mean it's common knowledge. The footage Tim watched didn't necessarily make it to national level. And if it wasn't too clear, there must have been one or two weeks of experts debating on TV whether they thought it was genuine or not, and whether Batman really exist or not. In the shadows, Batman would be less easy to see than Robin - maybe only a shape, and then it'd be hard to say it if was a hero or a thug or a ghost. And that's where the crazy theories about Robin and his Spirit of Shadows come in.
It seems so strange to imagine that to the general DCU, Robin doesn't have to be The Sidekick.