I have found the problem: I had a incorrectly configured workflow as part of the Logic App. What was odd is that there was no validation error when I was saving this workflow even when I saved it in the portal.
If you want to check whether you also have such an error, restart your Logic App, go to Application Insights, select the Transaction search and wait a few minutes for the logs of the restart to get ingested into AI.
Then, you can look for the startup logs of your Logic App. As part of these, you will find any information about possible errors in your workflow configuration.