I was able to get FaceTime to work in high res but it was super annoying. Basically open FaceTime and then open the Logitech settings app, if you switch that to widescreen you'll see the FaceTime camera jump to a slightly higher res version. THEN I tried opening OBS or running the website that OP listed for camera tests and the FaceTime camera jumped to 4K.I have an almost identical set up to OKiRun.. so that's cool.
I have the Logitech 4k Pro camera. Using the longer cable, I connected the camera to one of my external thunderbolt 3 raids, rather than my Mac Pro. My Mac Pro's ports are spoken for basically. With this, I was able to open Quicktime Player, go to File / New Movie Recording.. and then on the actual video pop up window, click the down arrow beside the record button to switch quality from high (1920x1080) to Maximum. It then switched to 4k mode.
This all said.. in Facetime, wow. Potato camera. Low res and then some, no idea why.
Seems like FaceTime itself isn't asking for the highest res from the camera, but when an other app does it'll use that feed instead.