I pulled the trigger on a GR IIIx about a year ago. It's mind bogglingly amazing and several orders of magnitude better than anything my iPhone 13 Pro nor my Pixel 6 XL can muster. It goes everywhere my smartphone does as it is very pocketable.
The GR III/IIIx does not have HDMI out, however it does support the DisplayPort protocol where you can solve video output in various ways with adapters, do note the max video resolution output is 1080p. The main problem is that eventually it will overheat and turn off (within minutes). Video was and is not a strong suit with Pentax/Ricoh products unfortunately.