I can understand that a zoom lens with a maximum aperture wider than f/2.8 would be difficult and costly, but it seems like something pros would kill for. Surely it can't be more expensive (if its possible) than some of the $10k+ lenses and they exist to sell in limited quantities. Is it just near impossible or is there another reason we don't see zoom lenses faster than f/2.8?
As for why these aren't common, consider that this particular lens weighs 11 pounds and costs about $50,000US.
Going even more extreme would be the Optimo 24-290, which is f/2.5 (T2.8), and constant aperture across that entire range. It weighs 24 pounds, and though I don't know a price on this one, I think: "more than most houses I've lived in" would be a safe guess.
Yes, you undoubtedly could build an f/2 zoom that was smaller, lighter, and less expensive, especially if you restricted it to APS-C. Nonetheless, this may give some idea of the direction really fast zooms would go. Yes, they can be considerably more than expensive than $10K, and they can be unreasonably large and heavy as well.
From an optical viewpoint, it largely comes down to correcting aberrations. Just for example, for any particular design spherical aberration grows approximately quadratically with aperture.
Along with that, you run into size and weight problems: to get one stop faster, you multiply the diameter by ~1.4. That doubles the area, which multiplies the volume of each element by 2.8. With each element weighing about three times as much, the mechanical parts to mount those elements end up bigger and heavier as well.
So, let's consider one of the most popular fast zoom categories: the 70-200 f/2.8. Most current ones way about 3 pounds. Based on simple geometry, scaling up to f/2 should increase the weight to about 9 pounds. At 3 pounds, you're right at the border between hand holding and really wishing for a monopod. At 9 pounds, most people pretty nearly need a tripod for any more than one or two shots at a time.
Along with that, 9 pound lenses just don't sell in large quantities. Reasonably dedicated photographers buy a lot of 3 pound lenses (including the aforementioned 70-200/2.8). When you get to the 5-6 pound lenses like a 300/2.8, the quantity has already dropped a lot -- only a few of the most dedicated will even consider them. Going up a step from there (e.g., to a 400/2.8) the number drops precipitously.
Let me try to put that drop into perspective. When I go to one of my kids' sporting events, chances are pretty good at least three or four of the parents will have a 70-200/2.8, or something around the same size/weight (and at a larger event, I might easily see a dozen).
For 300/2.8, that drops a lot. On any given weekend at my local photo-friendly wildlife refuge, I might or might not see one. During mating season (for example) seeing two or three wouldn't be all that unusual.
In twenty years of shooting, I could probably just about count on my fingers the total number of times I've seen people out shooting a 400/2.8, 500/4, 600/4, etc.
Based just on size and weight, a 70-200 f/2 would be pretty much in that last category -- so rare it would hardly be used even if somebody put it in their catalog.
Zoom/varifocal lenses with larger relative apertures do become more practical (and common) when dealing with shorter focal lengths and/or only needing to cover smaller sensors. Olympus has made f/2 lenses for micro-four thirds cameras for quite a while, Canon has announced (but as I write this, not yet released) a 28-70 f/2 for their new EOS r mount, and (perhaps craziest of all) the old Minolta 3x-1x Macro zoom, with a geometric aperture varying from f/1.7 to f/2.8 (but at 3x, the geometric f/1.7 aperture is effectively reduced to f/6.7--and you usually stop down from there, to try to get at least a tenth of a millimeter of depth of field...
Olympus makes f/2 zooms, but I disagree with another poster who said this was like f/4 because of the smaller sensor size.
Sure, the size of the sensor has an impact on ultimate image quality, particularly with depth-of-field effects, but you still get to shoot an f/2 in lighting that would be impossible with an f/4.
I don't see any postings sneering at the cited cine f/2 lenses, although they surely have a smaller "sensor" than 35mm full-frame, as well!