That video is clearly not encoded correctly. If it were the levels would match the background, given there is no actual HDR content visible in that video frame.
Anyway, even if the video was of a lovely nature scene in proper HDR, you might still find it jarring compared to the surrounding non-HDR desktop elements. I might too, depending on the specifics.
However, like I said, it's up to the browser to handle this.
One suggestion I saw mentioned by some browser devs was to make the default to tone map HDR if the page is not viewed in fullscreen mode, and switch to full HDR range if it is fullscreen.
Even if that doesn't become the default, it could be a behavior the browser could let the user select.
Actually I forgot about auto-HDR conversion of SDR videos which some operating systems do. So it might not be the video itself, but rather the OS and video driver ruining things in this case.
Just because we're in the infancy of wide HDR adoption and thus experience some niggling issues while software folks work out the kinks isn't a good reason to just wholesale forego the feature in such a crucial piece of infrastructure.
Sure, if you don't want HDR in the browser I do think there should be a browser option to let you achieve that. I don't want to force it on everyone out there.
Keep in mind the screenshot you showed is how things looked on my Windows until I changed the auto-HDR option. It wasn't the browser that did it, it was completely innocent.
It was just so long ago I completely forgot I had changed that OS configuration.
Anyway, even if the video was of a lovely nature scene in proper HDR, you might still find it jarring compared to the surrounding non-HDR desktop elements. I might too, depending on the specifics.
However, like I said, it's up to the browser to handle this.
One suggestion I saw mentioned by some browser devs was to make the default to tone map HDR if the page is not viewed in fullscreen mode, and switch to full HDR range if it is fullscreen.
Even if that doesn't become the default, it could be a behavior the browser could let the user select.