Why does Azure have this ridiculous behavior ? If the client encounters an error, I cannot ask them to repeat it after I turn on the registration again.
This is true if you use the default file system for logging. If you set up a vault account and then enable logging, you can choose whether the logs will be sent to the file system or vault.
Take a look at this article in the "How to enable diagnostics" section.
https://azure.microsoft.com/en-us/documentation/articles/web-sites-enable-diagnostic-log/