Just one little nit to start: the videos you see of the liftoffs taken from outside the LM weren’t “filmed” — they were captured on a video camera and broadcast back to Earth. There was no film involved.*

On the first three Moon landings (Apollo 11, 12, and 14) the video camera depended on the Lunar Module for electrical power, and on the LM’s transmitter and antenna to get the signal back to Earth. In this photo of the Apollo 11 TV camera on the Moon, you can see the cable that runs between the TV camera and the Lunar Module, carrying electrical power and the video signal:

The transmitter that sent the video back to Earth was in the LM’s ascent stage, the top portion that contained the crew cabin and which lifted off the Moon and took the astronauts to lunar orbit. Prior to liftoff, all electrical connections between the LM’s ascent stage and the outside equipment were severed, so the camera had no way to get a signal back to Earth.

What changed on the last three Moon landings was the presence of the lunar rover (Lunar Roving Vehicle, or LRV). The LRV had its own transmitter, antenna, and batteries. The TV camera was mounted on the LRV, and was remote-controlled by Mission Control from Earth. It could continue to operate and communicate with Earth as long as its batteries held out, and didn’t depend on the presence of the Lunar Module.

(Apollo 17 LRV)

Once you had a TV camera that was remote-controlled and could continue to operate during the LM’s liftoff, how could you possibly not try to get video of the liftoff?

*I get it that people often use the word “film” when they mean “record video”. I still talk about “taping” a TV program even though I use a DVR that has no magnetic tape in it. The only reason I think it’s worth mentioning that there was no film involved is because of the ridiculous Moon landing hoaxnuts.