Nick, it looks like there are other ways of doing this, but, like you said, there are some limitations:
Extract Data—ArcGIS Online Help | ArcGIS
One of these has to happen:
- You own the features.
- You are an administrator for your ArcGIS Online organization.
- You aren't the hosted feature layer owner or the administrator, but the owner or administrator has configured the hosted feature layer to allow others to export the data.
In any case, it sounds like Open Data will get you what you need.