Is it possible to send a triggered send transactional email without adding that contact on the All Subscribers list?
All Subscribers is Marketing Clouds master subscription list that manages the contacts subscription status.
There's no method of using a Triggered Send without it adding the contact to All Subscribers - Marketing Cloud needs the contact in All Subscribers to manage the send. When you configure your TSD, you can add the contact to a list, but even then the contact of a list will also add the contact to All Subscribers
This is the same functionality as when you have a contact in a data extension and you then send an email to that DE - the contact will automatically be added to All Subscribers.