It's not so much that CRTs are faster than LCD.
It's that old consoles (up until the Sega Saturn, Nintendo 64, Sony Playstation era) didn't draw to frame buffers. They didn't have enough RAM to have a real frame buffer.
You configured the graphics chip what you wanted it to draw, and as the scanline scanned across the screen, it outputted the color that was supposed to be output depending on the background/sprite/palette settings. You could change where the sprite is halfway across the scanline, and it might immediately, in the middle of drawing a scanline, output a different color. There was no delays anywhere in the system, only the propagation delay of electricity in wires and transistors. The various HW registers that controlled what and where the sprites were were connected via a small pile of non-clocked digital logic gates to a DAC to the analog output pins of the graphics chip, which was connected to the electron guns directly if you used component video or via a simple, no-latency analog circuit if you had composite video.
These days, you program the graphics card to draw what you want it to draw. Once you've programmed everything for it to draw, it draws that into the backbuffer. Once the current frame has been sent to the display, the backbuffer is swapped with the frontbuffer. When the next frame begin is sent to the display, your changes finally go out across the wire. Depending on the display (see especially motion adapting TV screens) there might be more delay. Then whatever changes you make are finally displayed on the screen.
It's kinda weird. Old video game consoles (up until the SNES and Genesis) had extremely low latency. And that's been gone for 25 years. Not only is it gone, but it will likely be gone forever- we don't even make the technology anymore to show the new generation what it was like. On the one hand, the new technology is "better"; there's no way to do today's advanced graphics without a deep drawing pipeline that outputs to a frame buffer. But it's also somehow worse. We can make it less bad with technology like 144Hz and Freesync, but the old era is gone.