Internet archive seems to only have the layout of things saved (snapshot) but not the actual contents of the page.
This does not contain any of the information, just what it looked like when it was saved I believe. I am wondering if I am wrong on this.
Nope. The content is served through Steam, not on the page. You'll have to find someone who mirrored it if it's been taken down.
Edit: however, if it's still available in the Steam CDN system, you may still be able to download it with a tool like this: https://github.com/SteamRE/DepotDownloader
Do you know how to use it?
I have no idea how to find a manifest or a depot
The only required parameter is the appid, which is in the page URL. I don't really know what the other IDs are, but you can mess around and see if it works.
You need a Steam account that owns the app (aka game) you want to download Quote from this thread
I would theoretically need to own the DLC for it to download. You can only download things you own on your account.