If you are using the Lightning Email Builder, you likely want to save your email templates in a public folder. The public folder will allow anyone in your team to access the email templates. (Saving the email templates in the Private folder means only you will have access.)
But sometimes, you can’t see “Public Email Templates” as a choice.
If this happens, you need to update the Permission Set.
1. Go to Use CMS with Email Templates and Email Content permission set
2. Go to System Permissions