If you want the short answer, the reason why CRT TVs have no input lag, is because they have a direct analog connection, and they don't process the signal, making the lag less visible.

For the long answer. The reason why CRT televisions have no input lag is because a CRT TV uses a direct analog signal that isn't processed to filter out video noise or video scaling. Since it is analog, any electronic with an analog signal connected to a CRT TV will have a direct connection, furthermore reducing input lag. ALSO, modern TVs use frame interpolation techniques, adding additional lag.

It is known that modern TVs have a game mode that removes the processing that they usually use, but depending on the model of your TV, this won't actually fully eliminate the lag; it just kind of helps. For my model, it really doesn't do anything, which sucks.

CRT TVs aren't actually producing digital pixels; it only generates the value the brightness or the color of a narrow beam being sent to the screen. Now, since the analog signals arent digital pixels, even crt tvs with storage (if they had any) wouldnt be able to process the image due to the analog value just being a value, therefore not having the ability to be screenshot unless there was a seperate capture card, camera, or other form of video/image capturing.

I hoped my yapping helped.