Yes you’re absolutely right 6500K is the standard for CRT displays and more generally. However 5000K is seen as neutral white (although opinions vary) and this is for the camera i.e the camera is neither adding nor removing warmth.
We could set the camera warmth to 6500K but then you’d be effectively adding 1500K (i.e 6500K - 5000K if you accept 5000K as the neutral white point) onto the TVs temperature in the photos.
However really all we care about is that we know what settings a photo is taken with so we can adjust if needed locally.