How can I recover data from a website after shutdown?

Solution 1:

You are almost assuredly correct that a database ran in the backend. Since the site was shutdown, without a doubt, the database and server side code that ran it is completely gone. The page you are seeing is likely the default HTML output of the server side code and database. Archive.org can only copy HTML output of a website. It has no ability to see what is creating that output. In essence, it has a "photograph" of the front page and that is it. In all reality, there is nothing else that you will be able to recover, other than what you see.

Solution 2:

How can I recover data from a website after shutdown?

YOU probably cannot.

As you've found out, the Wayback Machine has limitations on what it is able to cache from the web.

If you didn't cache it yourself while the site was operational then you would have to find another person that has archived what you seek.

Is there a way to bypass the archived page entirely by accessing the database directly?

If the database still exists then you would have to contact the owner to gain access. It is very unlikely that it is freely available for access over the Internet.

Usually companies will archive data of sunsetted products and only resurrect the data if there is a legal requirement to do so or if a new project could benefit from the old data in some way. I doubt Twitch cares about your personal interests but you could try reaching out to them.

If someone has a copy of the database then they very likely obtained it illegally.