The API Steam downloader (and similar web pages and tools) uses is a web-based API - it never had anything to do with the game itself. This is nothing to do with Steam Downloader and the API, and everything to do with the publisher of the game not enabling those who don't own the game on Steam to download workshop mods. In the past the steam workshop used to use a different system where the game itself would download mods using an API, which I think is what that steam workshop downloader is trying to use. It all goes through the steam client now.
'I think it's not actually possible to use a downloader for newer steam workshop stuff, because it uses the same download depot system for mods as for games. Guess GOG users are not allowed to use mods for that game. 'The game that this item belongs too does not allow downloading of its items' on for some games like Hat In Time, you're fucked
