I don't do formal HDR work but I've done lots of HDR HLG tests in FCP using an Apple Studio monitor and LG 5k monitor, both connected via Thunderbolt to my M1 Ultra Mac Studio. I've also tested on my M1 Max MacBook Pro 16. I've also uploaded HDR tests to Youtube and Vimeo and tested streaming playback on the above machine.
In general it works OK but Safari support for Youtube HDR is sketchy. It works well on Vimeo and MacOS Firefox works better on Youtube for streaming HDR content.
With those exceptions, in general I haven't seen any problems on recent prior or current versions of MacOS and FCP. But I don't have any machines which connect to monitors via HDMI.
I think there are some limitations on HDMI cable and endpoint versions for full support of 4k/59.94 Rec.2020 HLG with Dolby Vision 8.4 (what FCP exports). I don't recall any of the above posts stating what resolution and frame rate were being used.
It would be a good test to try 1080p at 23.98, 25.0 or 29.97 and see if that works using Rec.2020 HLG with Dolby Vision 8.4. That lowers the bandwidth a lot, which brings the scenario inside more specs for cables and endpoints. If making that change works better, that tends to imply it's a cable or endpoint issue.
When using HDR output from an Apple computer to an external monitor, a key issue is what display profile is used. If the display profile is not configured for HDR or one of the new presets like XDR P3-1600 nits, it won't look right. The newer Apple monitors have presets and the older ones and non-Apple monitors normally use ICC profiles. Which one is in use is shown in System Settings>Displays.
On any Mac with either built-in monitors like XDR MacBook Pros or with external monitors, you can test HDR by going to Youtube and searching for "HDR channel" and trying to stream those with Safari and Firefox. If you right-click and select "Stats for Nerds" that should indicate 'Color HLG / bt2020.' If it does not, there is a problem independent of FCP.
iPhones 12 and later can record HDR using 10-bit Rec.2020 HLG Dolby Vision. That does not have to be ProRes. If you enable HDR recording on the iPhone, you can upload that directly to Youtube or play the local file on an HDR-capable Mac -- without editing. That is another way to test whether your computer playback system can handle local or streaming HDR playback. Only if that works on the given computer and monitor is there any need to pursue whether you can edit and grade that using FCP: Adjust HDR camera settings on iPhone - Apple Support