Hey @Nathan88
I assume you're using the database server's fully qualified domain name (FQDN) <servername>.mysql.database.azure.com
in your AZURE_MYSQL_HOST. When you created the private link, there should've been a DNS CNAME record that resolves the FQDN to private link endpoint, so that's the first thing I would check.
If the DNS records are correct, then you can review this troubleshooting guide to help isolate the issue for your configuration. The Diagnose and solve problems blade has a Network troubleshooter that essentially follows this guide that you can use to assist you. If you aren't able to resolve your issue after reviewing the guide, please share any errors you received post troubleshooting.