Transactional Email API: TriggeredSend definition not enabled for this route

You cannot create a triggered send definition with the UI if you are using the Transactional Messaging API. You need to create your definition via the REST API in order for this to work.

Adding here as I spent a few hours trying to figure this same issue out last night. As this answer states, you do need to set up the send definition using REST API. You cannot use the UI to set up the send definition for transaction API driven emails. It's called out here: https://developer.salesforce.com/docs/atlas.en-us.noversion.mc-apis.meta/mc-apis/transactional-messaging-api.htm

I realized my confusion (perhaps yours too) is that there is a lot of documentation out there for 'triggered sends' that show people setting up send definitions in the UI and then executing sends via API. This is different from the Transactional Message API (which doesn't allow for any UI creation or editing of the send definition). This is a huge miss if you ask me as in a business setting, there is a need for business people to change content as needed and this is not possible without a UI option.