Is there a list available of which restricted applications each duty position needs to be able to do their job?
Yes. If you're using the "Preview" version of eServices, then in the left-hand column there's an item called Reports, and in the popup menu under Other Resources there's a link called Applications by Duty Assignment. Tells you just what you're looking for.
It's in the footer of old eServices called Applications By Duty Assignment. https://www.capnhq.gov/CAP.ReportingModule.Web/reports.aspx?SID=61B01969-4DC0-4719-9A3F-E092572F547A&FID=PDF&RID=121 should take you there.