I am not aware of any "uber / taxi" camera that don't have a screen, sound strange that "NO screen or display viewable or able to become viewable with the push of a button "
While we Danes do have a law against video or TV footage going on a screen the driver can see ( for good reasons ) then NAV screens are of course okay and as are large headunit screens though they can playback video and often have a TV tuner that could come live with a push of a button.
I don't think we have anything special for dashcams that admittedly are a "video" feed for the duration the screen are on ( variable shut down time normally in dashcams ) but it will of course come on with a push of any button.
We Danes have licensing for Taxi drivers, but nothing for things like Uber and so on, and hence those services cant operate legally here, but while some / many taxis have CCTV i am not aware of anything like your screen "issue"
The Taxis here normally use expensive PRO systems with no screen or option for driver to interact with it, but i have seen several driving instructors having consumer grade dashcams in their cars.
For the forward facing view you can find a camera with no screen, but for the cabin side ( which i find just as important as a forward facing camera in a taxi / uber ) you don't find any single or dual channel cameras without a screen, and not least IR light to aid the cabin camera at night as the car cabin are normally too dark for a unaided camera to be usable.
Sound like the Scots ( too ) have lost some of the good old skull braking claymore swinging spirit.