I'm familair with the constriants, everything looks like it should send the final email or an error email.

The problem is that if you are logged on as you it will connect as you and send an email as though its you, which isnt the same.

Thats why I think you have a problem with the SQLAgent connecting to the DBMail.

I think you might need to set up the Account SQLAgent runs under to be a member of the role DatabaseMailUserRole in the MSDB database.

Grant the permission and restart the SQLAgent.

Theres a simple way to check create a simple 1 step job that does a TSql Select 1 from a database

Set up the Notification at the job level to send an email on success and execute this package through the Activity Monitor, dont run it through the jobs, you should receive an email.

If not check the Log files under SQLAgent and you should see an error being raised in there.