Lightning Email Templates Are Unavailable For Use On Outbound Emails

When the email template created doesn't show up under 'All Email Templates' on outbound emails,  it is because of the entity type. ‌For the global Lightning Email template, the template must have the entity type of Contact, Lead, or nothing.
 

Change Related Entity Type Value of your Email Template

1. Go to App Launcher and search for Email Templates.
2. Go to All Folders, then select the folder where you created the template.
3. Edit the email template and change the 'Related Entity Type' field to -- None --.
4. Click Save.