https://webcompat.com/ to report. (Also in the menu: "Report broken site".)
Who are these people? What can they do?
They are the collective main browser makers, including Mozilla. One thing Mozilla can (and does) do is insert site-specific workarounds (e.g. changing the User Agent), and/or work with the website to ensure compatibility.
Yeah, it's right next to the wall of e-commerce sites that give you a single checkbox for "send me order information and tracking updates along with occasional promos"
I think the streaming with Xbox Cloud Gaming does not work on Firefox and requires a Chromium based or Safari browser. I believe the reason is because Firefox does not have an API for streaming games? Or the current implementation is not compatible with it. I read stuff like that in the past couple years and don't actually know why Microsoft does not support Firefox. So that would be a thing to put on a WOS: Wall Of Shame.
I rarely run into such sites, when I do I leave.
I've never run into one that I must use.
A place to discuss the news and latest developments on the open-source browser Firefox