You seem to have a serious misunderstanding of cinema lenses. Primes are almost always cheaper to cover the same range than zoom lenses. Comparing prime lenses that cost thousands of dollars a piece to a $1000 zoom lens is not a valid comparison.
I have a $2400 Canon 24-70 f/2.8 II. It is the single best zoom lens Canon has ever made for under $50,000, yet it is only comparable in sharpness to the $300 - $400 f/1.4 lenses and barely exceeds the sharpness of the $100 f/1.8 primes.
Additionally, sharpness is only one factor to consider, you also have speed. Long exposures aren't an option with video, so if you want good low light performance, you need a fast lens. Prime lenses are much, much faster than zooms and thus provide much better low light performance while keeping shutter speeds quick enough to avoid too much motion blur.
Finally, even high end photography lenses are not parafocal, meaning that as you change the focal length, the focus shifts as well, making them undesirable for any kind of push or pull shot (though you can work around it by following focus carefully.)
The cinema lenses help with this problem (though they don't solve it entirely), however, the Canon prime lenses are $20,000 a piece while the zoom lenses are $55,000 a piece the last time I looked at their pricing.
So in short, primes are the cheaper and higher quality option on a budget. You only really want zoom lenses when you need the flexibility on the fly or need to execute a pull or push shot. Given the difficultly of pulling off a good pull or push with photography lenses, it isn't that common to bother with them unless the camera is also used for photography.