Thank you for your feedback. We are currently in public preview of static website hosting for Azure Storage to enable this scenario. Check out the blog post here for more details: https://azure.microsoft.com/en-us/blog/azure-storage-static-web-hosting-public-preview. The feature set includes support for default documents and custom error documents for HTTP status code 404.
For any further questions, or to discuss your specific scenario, send us an email at firstname.lastname@example.org.Jason Turner commented
This would include the original URL as well so the above mentioned files can be tracked. It would also allow for timing out of SAS tokens and re-authenticating/re-issuing.
At the moment we have to deploy a separate app to be able to do this and it's just a pain to manage...