There are many time-lapse DSLR videos out with great star fields moving across the sky per the rotation of the earth and these are very popular.
Here are a few examples:
...this has the moon, but notice how you can't see any detail of the moon's surface as it is overexposed to capture star light:
The above are for the most part are realistic and shot with DSLR cameras but in time-lapse which can allow for the needed exposure time (both dragging the shutter and opening the lens).
In Hollywood feature films it would be more likely and practical for the editors to composite star fields as shot separately or create in animation as this allows for more control of the total picture with predictable results.
In big productions there is less of a chance that they would try to shoot stars in real time with actors or other subjects simply because the difference in exposing stars is huge compared to that of a lit set. Just as you mentioned that you don't see a detailed moon shot with stars in the background because the moon is like 12 EV units while the stars are like near zero, that's like 12 or more stops difference.
For the present, to get a detailed moon surface shot with a star background would require a composite.
To be clear, most feature narrative films will not be real stars on their own, possibly composites with the main scene or animation or other effects while many independent cinematographers will use time-lapse that is real however sped up in time.