The behavior that controls private group accessibility by unauthenticated users is not modifiable in a supportable way. Moreover, publicly displaying even just simple lists of names+links to private *unlisted* groups leaks data about them that would not otherwise be exposed by the platform, itself.
However, in this scenario, I wonder if it would be possible for you to instead have a single content page on a public site which hosts your custom widgets. The external group listing links could all link to the same page with data about the intended group embedded within the querystring, and the custom page + widget could process that (and potentially redirect) in a similar manner as you are intending.