If clicking the button 'Send Test Email' ends up in an error message or the test email does not arrive, it is possible to debug the mail function and find out what is wrong. Follow these Joomla tutorials to get a detailed log, consisting of the SMTP connection dialogue and all the status codes. If the reason for the failure is a mystery, create a topic at the Joomla Support Forum and copy and paste the log file for the volunteer Joomla experts to study and give their advice.
How to debug SMTP mail in Joomla 4
How to debug SMTP mail in Joomla 5
Please note that Joomla 3 and Joomla 4 send the test email to the email address specified in the field 'From Email' in the Mail section of the Global Configuration,, a few lines up from the Send Test Mail button, whereas Joomla 5 and Joomla 6 send the test email to the email address of the logged-in Super User. That makes sense, because it allows you to test different email addresses, but remember to change your email address back after the test.