Why does my rule have no effect on a VBO list of unpublished nodes but does work for published nodes?
Anything about Rules that uses the Rules Scheduler, runs for sure with "Anonymous" authorizations. There are cases (Rules Actions) for which that auth is insufficient of course.

So a possible explanation for the cause of your issue, is that your unpublished nodes are not included in your VBO views result. To verify if this is indeed the reason that causes your issue, just check the views results you get by looking at the Views results without being logged in (so as Anonymous user). If those unpublished nodes are included anyway, verify if Anonymous users are authorised to delete those nodes.