How could this have been implemented in a way that obviates all of these show-stopper failings? Respect users' software freedom by offering users one or more programs to do the conversion jobs with reasonable default settings. Distribute complete corresponding source code alongside the rest of the site so as to let the user host an instance of the site on their own. Users so inclined can set this up for themselves and others, or use their software freedom to end up with a conversion service they don't need to guess about.
To those that are interested in understand out how the website works, the process goes as follows: the files are uploaded to the server and then uploaded to a private s3 bucket where the user will have an hour to download the converted content via a presigned url after which it will be permanently deleted. Furthermore, as soon as the file is converted, the original file is deleted from the server. In hindsight, I should have shipped an "about" page explaining this process.