I'm curious if the physics allow a company to build such a lens of course with good IQ, minimal color aberration, maximum sharpness, and little to none lens vignetting or edge softness
Theoretically one can do anything. It just takes more glass — and correspondingly, size, weight, and cost. A 300mm f/1.2 lens would need an apparent aperture of 250mm — which is almost 10 inches in diameter. And for extra complication, that needs to project an image circle which fits through the lens mount, which is probably around 2 inches. So that's a challenge, requiring even more glass. And all of that is going to cause more light loss, and all the other drawbacks of more transitions between lens elements. This would get expensive very quickly, and that's not even worrying about avoiding other trade-offs.
But okay, let's keep going. Canon actually makes (or has made) a special order 1200mm f/5.6 lens. That actually works out to the same aperture diameter as a 300mm f/1.4, so let's consider some of the specs of that lens as a ballpark for what we're looking at with your theoretical lens:
Since we're not aiming for such a large focal length, it wouldn't need to be quite that big, but you don't necessarily save much. Let's say half the size and weight — but the savings in price will be smaller.
And this lens isn't flawless in terms of image quality. It's got significant vignetting and isn't sharp wide open. If you want to fix that, consider doubling the size and weight and increasing the cost by an order of magnitude.
And finally, all of that is without considering the complication of zoom. For that, I guess we'd double the size and weight again, and raise the price by a another order of magnitude. If you really want that 17mm wide-angle, it's probably even more for all of those factors.
So, if you've got an extra $12,000,000 or so, you might think about it. Otherwise, you might instead spend $10,000 or so on a 300mm f/2.8 plus a camera with great high ISO performance, and be content.
Your question raises an interesting point as there are some lenses you really can't make, for example you'll never have a 50 f/0.2 as physics simply wont allow it.
However a 300 f/1.2 is merely an engineering problem (building a barrel to hold and move the giant lens elements required), after all a 300 f/2.0 lens was made and sold commercially by Nikon, there have been other extreme fast telephotos, for example Zeiss custom made a 1700mm f/4.0 medium format lens for a private customer, which dwarfs the Canon 1200 f/5.6L.
The zoom complicates things, a lot, as telephotos are pretty simple designs, but to get to 17mm requires a retrofocal design.
You almost certainly wouldn't want to have one of these lenses, except to put on display as the performance wouldn't be great and the weight would be enormous. Here's a mock up of what it would look like (made by splicing the Nikon 6mm f/2.8 fisheye on the front of the Nikon 300mm f/2.0):
I'm not convinced it's possible to make something covering such a wide zoom range, with a fast aperture, and of high quality. mattdm makes many good points and has a solid analysis, but one thing I notice that makes me question whether this is possible: The wide-angle and mid-range f2.8 zooms (from any manufacturer) have consistently improved with each generation. For example, Nikon went from a 35-70 to a 28-70 to a 24-70. Why didn't they initially create a 24-70 2.8? Surely there would have been solid demand for it in the 80s and 90s; I can't imagine any reason it would be considered "unstylish" to have such a range at the time -- I think it would have been a popular lens then, just as it is now. Similarly, wide angles have evolved from 20-35 to 17-35 to 14-24; note also that an wide-angle zoom didn't exist from Nikon until '93.
The 24-70 optical formula is more complex than the 35-70, but the math in creating optical formulas hasn't changed over the years. In other words, a 24-70 could have been designed in the 90s. Nikon's ED glass has seen more and more use and other advances like their Nano coating further help to improve image quality. So, my guess is that with the technology of the 90s, a 24-70 f2.8 couldn't be created with the required level of quality (whether the problem be in distortion, color, sharpness, etc, or any combination).
No doubt, a 17-300 f1.4 would be a giant, heavy lens with a very complex optical formula and a variety of elements moving to make everything focus and zoom. So it could be built, yes. But what would the quality be? Based on the wide-angle and mid-range zoom evolution, I suspect that today it could not be made.