It actually also struck me pretty hard when I see the video and first noticed that the stars are missing in all the apollo photos.
Naturally, a full dark sky resembles the night sky in our earth view. So naturally I also believe there should be a shiny beautiful starlet sky on moon. Upon this puzzle, I tried to search more achives from NASA including mars pictures. But then I realized that if NASA lied on Moon mission, it may lie on other missions too. So a single source won't be too convincing anyway. I tried China's ShenZhou mission pictures then. To my surprise ---- there are no stars either.
I see little point for the Chinese authority to lie --- esp. in exactly the same way as the American lies. So I flipped the doubt to my own thoughts.
Note that all the pictures are taken by normal optical cameras. These cameras have better lens, yet they still need the normal reflection of light to work. And all of them of course must focus on the spaceship or astronauts as these are the major objects to take ---- for the press view at least. (the other pictures without spaceship or any near objects, are focused on the stars themselves, and of course they have stars.) Without any atermosphere, the reflection of sun light from these near objects are extremely strong ---- our eyes or camera lens will automatically adjust to such condition and the "shining stars in the background" inevitably became too dim to show in either naked eye or normal optical lenses. Just like you can't see stars in a busy city district during the night.