As I pointed out in the original post, ScheduledTask works if you use Now in the timesec, but not if you use a date in date object or datelist format.
So that issue remains unresolved and ScheduledTask does not work as advertised with LocalSubmit.
However, it does appear to work with SessionSubmit, which is fine as a workaround, although it would be nice if WR would fix this problem in an upcoming release.