I only know of HDR existing as a photography edit, so I'm not sure how this is applied with respect to screen brightness.
HDR for video is somewhat different than photos and requires a wide color display with high brightness. HDR photos are typically made from multiple exposures overlapped to even out the lighting. This can give you a weird painted/fantasy look to the photo.
HDR in video is far truer to life where brighter parts of the picture are displayed accurately without blowing out to pure white. You’ll get video where a sun is so bright it almost hurts your eyes while also remaining orange and other parts of the picture can still be in a deep shadow. Some elements can be so bright (fire / or the sun / etc) than many HDR capable screens can only display that amount of brightness in short bursts or on only a part of the screen without overheating. There are various HDR standards right now with inexpensive 4K TVs claiming they have HDR but usually a poor version with weak brightness. Dolby Vision is considered the best HDR and that is what the new iPhone has.
All that said, not all 4K content is mastered for HDR and no 1080p video standard supports HDR. Netflix has some HDR content, but only if you pay extra for 4K streaming.
I’m a director and cinematographer and I’m not a fan of HDR. On a technical level it is a pain in the ass to deliver and varies too much right now by device/tv, if you can even find a way to get it onto your screen in the first place.
Creators also have to make sure content will still look good on non-HDR screens, so most of the picture does not take advantage of the additional brightness range. Sometimes the stuff that does is distractingly brighter than the rest of the picture. Flames in Mad Max Fury Road were so bright they suddenly looked like cgi or elements from a video game overlaid over a normal movie.
I’d rather leave the brightness / contrast ratio of HDR to cartoons/video games/and nature docs as all that brightness can make narrative movies lose a bit of their timeless magic. It’s just a little too real life for me.
Colors are represented much better with HDR in general (at 10 bit rather than 8) and that’s always a good thing, but a lot of HDR content is oversaturated for a wow factor rather than focusing on accuracy.