Hey there, Melt!
Thanks for reaching back out.
You are correct, using the remote app will not accomplish what you are trying to do.
The best option to run this automation would be using the Home app. This will allow you to create a custom automation that will allow the Apple TV to turn on. Keep in mind, the TV connected to the Apple TV would need to be HomeKit compatible for the TV to be turned on as well.
You can learn more about the Home app, what’s needed to use it, and how to get started. Once set up, you will create an automation within that app to do what you need: Create scenes and automations with the Home app - Apple Support
Kindest regards.