Looks like they offer a full data dump: https://dl.opensubtitles.org/addons/export/

I haven't actually tried to use it. So long as they offer a dump you could use I think it's entirely reasonable to require payment for the API.

I think it's reasonable to consider them morally open even if the web UI is closed source if the subtitles are open. It feels to me like a project selling a shiny hosted app around an open source project. I think in this sort of project the core "source" is the subtitles, not the web UI.

If OpenAI let you download their models but didn't provide the tooling you'd need to run them I think their name would be much more reasonable.