I tried this, but no definitely nothing to do with the log shipping configuration. It seems far more to be to do with alerts in general; I know this because I've set up a different alert on a custom error message, configured that to notify an operator, run a RAISERROR calling the alert and seeing that be sent to the SQL Server Error log but no alert fired.

So log shipping is a red herring; it's entirely to do with SQL Server not firing alerts when it should.

Any thoughts on where to look to resolve this?