Is there any way to access email templates from profiles other than that of a system administrator?
You can add it for users. There's a
Manage Public Lightning Email Templates system permission outlined in this help doc.
You'd add that permission to a permission set and then assign that to any non-admins who need to manage email templates.
Users need the Manage Public Lightning Email Templates permission to create, edit, and delete public templates. Users with this permission, as well as users with the Modify All Data permission or the System Administration profile, can edit and delete any public templates, including ones they don’t own.