Our Zabbix 3.2.3 server had accumulated about 50,000 alerts (probably through some temporary misconfiguration). At this point, nothing we could do would get the messages being sent.

Nothing was shown in the logs which helped even at debug level. It appeared that Zabbix successfully queued more alerts, but never sent any.

The solution in the end was to simply delete all records in the Alerts table with status = 0. However Zabbix should either properly handle a large alert queue or at least put severe error levels logs into the server log file.