Error: NOTE: Failed to notify 'operator' via email. when trying to send e-mail when job fails
As you allude to in your question, my issue was indeed the mail profile in agent - thanks for the tip.
Other tips that are mentioned at the bottom of this post include:
Ensure DB mail is enabled within the SQL Server Surface Area Configuration. Do this by right clicking the server name and selecting FACETS from the menu Select the 'Server Configuration' from Facet dropdown Ensure Facet Property of 'DatabaseMailEnabled' is True
Set up a Mail Profile within the Management Folder of your server
Within SQL Server Agent set up an Operator
Look in the properties of the SQL Server Agent (right-click properties) and under the Alert System page tick 'Enable mail profile' and select your default profile that you set up in step 2.
Restart the SQL Server Agent
Second restart of SQL Agent fixed the problem.