Please enable Logs and see try to simulate the Problem Management Event on which an EMail is generated. As you said issue seems to be in Generation of email in Notification Engine.
The filters that start the Notifications (i.e. PBM:PBIL:AssignInd_810_SetTag; note all of the applications have a set of xxxx_SetTag filters), make sure the User and their group are not creating, assigning or resolving the Problem record (or Incident, Change, Asset, etc). Which makes sense because if I am creating, assigning or resolving the Problem record, I do not need an email to let me know what I did.
Thus one has to create test user that is not a member of the Problem Manager group or assigned group in order to have the notifications to work.