Canon 24-70/2.8 doesn't have IS, either.
Only Tamron is offering a 24-70/2.8 with IS (they call it VC, instead).
Canon has announced a 24-70/4 IS.
IS adds a lot of cost, as well as some size, a little weight and more complexity. It's harder to maintain high image quality with the additional, moving optics of any sort of stabilization mechanism, and I think most would agree that a 24-70/2.8 from any manufacturer should strive to be one of their most premium lenses.
I use Canon 24-70/2.8 and never miss not having IS with it. I just don't need it on these focal lengths and wouldn't want to pay extra for it.
I'm a huge fan of IS on telephoto lenses... have been using 70-200, 300mm and 500mm with it for 10 or 12 years now. I also have some shorter lenses with it, but it wasn't really very much of a consideration for me, with focal lengths shorter than, say, 135mm. I wouldn't want to pay much extra for it, but will take it if it's included or there simply isn't any choice and I really need the focal lengths and other features the lens offers.
Recently Canon has started offering, for example, 24/2.8 USM IS, 28/2.8 USM IS and has announced 35/2 USM IS is coming. I'd love to have a 24/2.8 USM or 35/2 USM, but not at the $800+ US price tags they have on these, a lot of which is due to the addition of IS.
I really don't know about video... if IS helps or is a problem. If using an IS Lens and panning with moving subjects, you see IS "working" at times... moving the image in the viewfinder. Would that be a problem with video? I don't know, I simply don't shoot video with my DSLRs.
There are other techniques to manage steady-enough shots with slower shutter speeds, without any sort of stabilization. Rest your elbows on a table.Or lean against a wall. Or just take extra shots, a short burst, one of them is likely to come out plenty sharp.
24-70/2.8L at 63mm on 5D Mark II, at 1/30 handheld (outdoor shade with polarizing filter and ISO 3200, no support, just took a series of shots, some of which were fine)...
50mm lens on crop camera at 1/30 handheld (my elbows resting on a table)...
85mm lens on crop camera at 1/80, handheld (leaning against a wall, the only light source in a darkened room was from the projector, it's noisy because I was pushing ISO pretty high on a 30D to get this shot)
100mm lens on crop camera at 1/60 handheld (window daylight only, handheld and not using any support, just took plenty of shots)...
180mm lens on full frame (film) camera... unknown shutter speed, but since I use ISO 50 and 100 film, it wasn't very fast... Avail. daylight outdoors, handheld. Well, okay, camera and lens are actually resting on the ground.
Of course, you can use a monopod or a tripod. Another trick is a 5 or 6 foot long piece of 1/4 or 5/16" cord/rope with a 1/4x20 bolt tied to one end, screwed into the bottom of your camera. Let the rope drop and step on it to make it taut. It's surprising how much this helps. Plus you can roll up the rope and stick the whole thing in a coat pocket!
Keep in mind, IS or OS or VC can only help keep your camera and lens steady (and you still need to use good techniques to get the best out of it). Stabilization of any sort can't help stop subject movement. When you get into these low shutter speeds, your subject has to remain pretty still. Even a light breeze on a flower or leaf can cause problems.
Yes, it's easier to handhold lenses on full frame such as 6D, than on a crop camera... However when you get to 1/30, 1/15 shutter speeds you start to get in the range where mirror slap can cause enough shake to start showing up in images, and that's worse with the larger mirror and shutter of a full frame camera than it is with the smaller ones in a crop camera. You start to get into speeds where you might want to consider using mirror lockup with most cameras starting around 1/30 or slower, down to around 1 or 2 seconds, beyond which any mirror slap is only a small percentage of the overall exposure and fairly insignificant.