Here is an answer I got from Kentico Support directly.
I believe the issue here is related to the global events that normally trigger automation processes not being included in the more limited version of the API that is available for external applications.
However, I think it should still be possible to trigger this automation process from your external app, using code that explicitly does so, like this: https://docs.xperience.io/api11/on-line-marketing/marketing-automation#Marketingautomation-Startinganautomationprocessforacontact
Hope that helps you!