So it sounds like you're trying to figure out a way to know when FusionAuth has completed startup.
The webhook is one way to do that. As you said, you can set up a kickstart event webhook within a kickstart file.
Another way is to poll for a known value, such as a non-default tenant that you know your setup has added. That may be a simpler solution for you.
Either way, you'll have to write some code to kick off the testing once you receive a signal that FusionAuth is ready.
I personally would have preferred if there was an API endpoint I could query is to whether or not the kickstart has completed successfully, but instead we were given a webhook and I'm not quite sure how I should be using it.
You'll need to write a webhook receiver that will kick off your tests (or whatever the next step of your testing setup is). I'm not quite sure how do that in one github action, but it should be pretty easy to split up a github action into two actions, a setup one (where you set everything up, including FusionAuth) and a test action (which you kick off in response to the FusionAuth webhook firing).
I think you'd want the workflow_call event: https://docs.github.com/en/actions/using-workflows/events-that-trigger-workflows#workflow_call