The deploy seems to be dependent on AzureWebJobsStorage, while it now not the case that it is always set. Using managed identity one can use AzureWebJobsStorage__accountName and provide only the name.
Using RBAC authentication for deploy in this action one can have access to that storage account so no need for connection string.
I am not sure if this is a bug or a feature request, but it is nice to have this working and it is hopefully not that complicated to do.