To answer the question 'why would you want to record CRT screens in 4K', to capture the magical 'lite-brite' look of retro games on retro equipment. Instead of chunky, hard-edged blocks of solid color that you get playing old videogames on modern LED / OLED screens, you got smooth, blended colors as the original game developers intended. So a 16 color game, looked like 1,000 colors thanks to blending.
To answer the question of what camera to use... I personally use two Panasonic Lumix FZ-80 cameras...
https://amzn.to/3ynwMlp and the results (once I tweak the settings) bring a tear to my nostalgic eye! With the right settings, you'll seldom get a miscolored frame... here are two example videos that I shot with this camera off of two crt type screen.
Recorded in 4K from Tandy TGA monitor - pay attention to the slow motion/zoomed in replays after each death animation. I was amazed that YouTube preserved the CRT pixels so clearly at that zoom level.
Recorded in 4K from Sony Wega CRT TV
Recorded all the zoomed in footage in 4K in order to highlight the CRT pixels.
One caveat with this camera is that it can only record 4K for 15 minutes at a time. After 15 minutes, it autostops and you have to simply hit record again for the next 15. Hope this helps, and I can provide more info regarding the best settings for this camera
https://amzn.to/3ynwMlp if you go that route. Good luck!