I might be missing something, but wouldn't the point be that it's not the log writer's responsibility to do this, but rather some other service that consumes the unparsed output and sends the notification?
I think 20 years ago it would have been slammed as overkill to have a separate process just to send logs.
Even now actually, there is a flurry of libs/gems to send events/logs/analytics to a remote server (Datadog, NewRelic, Slack…)from the application itself. It’s usually not directly coupled with the logger, but it’s not far.
You can eat your pizza slice whichever way you want.
Arguably this would be useful for critical events.