The latest tagged version of the Data API Builder Docker image as of this post (0.8.44-rc) might be causing the issues.
I was running into similar issues while trying to deploy to Azure Container Apps using the sample scripts found at https://github.com/Azure/data-api-builder/tree/main/samples/azure, and changing the container image tag from "latest" to "0.7.6" (the most recent stable version as of this post) seemed to resolve the issue.
I'm not sure if it's related, but based on the sample scripts the file share should also be in ReadWrite mode instead of ReadOnly.