To understand what's wrong, I have activated the debug on the production server, just to see what happens.
This is dangerous as the Ignition page may leak sensitive information. Read the bolded part of the second paragraph here.
How is it possible to see the real error in place of the HTML page, like it does in local dev ?
The page should never be used in production. If there's nothing in Laravel's log files, then maybe the framework fails to boot and you need to check the web server's log files.