Can the alert go to Slack or Discord instead of email?

Yes, as well as the notice and the email. Under Settings → When it does not arrive, switch on Also post an alert to Slack, Discord or a web address and paste a Slack incoming webhook or Discord webhook address. Any other https:// address on the internet gets the details as JSON. An address WordPress will never post to (one on a private network, or on a port other than 80, 443 and 8080) is not saved, and the screen says why. It is posted from your site's own server, at most once a day, so it still reaches you when email itself has stopped. Send a test → Post a test alert checks it straight away.

The alert leaves out the address the failed message was for unless you switch on Include the address the message was for. The webhook address is stored scrambled and is left out of an exported settings file.