web.config not writable on Azure
You should not modify the web.config in a way which could cause problem to your application. What happens is that whenever the web.config is change it triggers a notification about this change which cause your IIS host process to recycle. After this notification IIS host process restarts by reload the updated/modified web.config and if you have made wrong modification your application will not run properly.
In your situation when you suggested web.config is "gone" it is actually not gone, it is there but not visible in your FileZilla due to some ACL issue (not sure what the reason for it) as I have seen same issue. Trying creating a new FTP session and see if the web.config is visible again.
Another option for you is to download the full site locally and then after editing locally you can publish it back with updated content and the site will be back as expected.