No it doesn't! Not for me anyway.

I assume it works for you?

I've attempted "Cookie Declaration" with three different browsers and had no luck.

Serren, I'm not sure what XHR is, and the description on the page makes it sound like this is a complicated program that will often tend to make websites not work well at all, but I can give it a try.

Yes, it pops up a large overview version of the previous consent box, listing all the various cookies. Clicking on "Change your consent" then pops up the aforementioned box, allowing to select or deselect various cookie categories (other than Necessary and Unclassified).

I don't know why there are Twitch cookies in the Necessary category. I don't know any part of the site that uses Twitch content.
You could potentially end up reproducing the same issue, if you configure each browser the same or install the same add-ons. Not that I'm convinced this isn't a site problem.
For troubleshooting purposes, if you use Firefox, type about:profiles into the address bar. Create a new profile there, click the "Set as default profile" button below your original profile, then click the "Launch profile in new browser" button below the newly created profile. You now have a factory-default state of the browser and can check if the Cookie Declaration link on the www.gog.com homepage works.
If you use Chrome, you should have a Profiles button the the top right corner of the toolbar that allows you to accomplish the same.

Yep. It was my blockers. Disabling them (temporarily) allows me to set my cookie preferences. Thanks.

Thanks for the advice.