Hi @FrederikL ,
Since your function is running in consumption plan, there is no option of Always on. From your observations, it appears that this issue might be caused due to improper function host startup. In order to check the function host startup issues, I would suggest you to run the "Function app down or reporting errors" detector present in the Diagnose and solve problems blade in Function portal.
You can also check the Kusto logs of the "Traces" table present in the Log analytics workspace (it is present in the Application insights linked with the Function app). This table would provide logs about the function startup.
Hope this helps! Feel free to reach out to me if you have any queries or concerns.