Why is ansible notify not working?

Handlers will only be fired for tasks which report a changed state. In the output of your play you can see the status is ok, which in this case means no new packages have been installed or updated (since state=latest)

So it would work, if any of the packages would not have been installed already.

Though, instead of state started I would use state restarted, since you're always upgrading to the latest packages in your task (state=latest) If a package gets updated, you also should restart the service.