If you stream or upload game footage ~~before it’s official release date~~, they have full rights to shut you down

The timing of it speaks to what kind of damages they might be seeking, but it doesn't change the overall principle of it. Streaming or uploading video footage makes a derivative work or a partial copy, and I think that's pretty well established by now, regardless of whether it's done before release, right after release, or 20 years later.

I know, people do it all the time, anyway. Well, there are "fair use" cases where copyright law can get a little more lenient (like if you're giving a tutorial, or the footage relates to some academic commentary), but most of the video game footage you see online isn't there because it's legal, but because nobody really cares about it (or is suffering any damage from it). In principle it's still not legal.