Historically speaking, I always found that Basiccolor Input worked well as a generall-purpose camera profiler - but you really should use it with a Colorchecker DC or equivalent. There is also, I think free software from Xrite, and some software from Adobe that determines DCP primaries and even lets you edit them. And there is even a one-shot measuring devices from Image Engineering.
Of course, I would agree that DCamProf is a serious effort - but is it really necessary? 
There is only really ONE sensor per major camera generation at the moment, the Sony one, and as you argue so eloquently it would be enough that somebody measure the primaries or spectral response and publish them once and for all. Tone curves etc are a different story.
Matching the color, making a colorimetric profile, is a mathematical task. It has some challenges, but the established science is there, you can be blind and make an excellent implementation. This is what you need when you make a profile for a scanner or a camera for reproduction (which essentially is using a camera like a scanner). For general-purpose photography you need a tone curve to mimic the higher dynamic range in the real world, and as soon as you apply a tone curve you no longer have any standard color science models that can help you. You need to use your eyes and invent from scratch, which is what the manufacturers have done since the start and come to very different solutions.
The free software from X-Rite has no tone operator, it just slaps a Adobe tone curve on top of a colorimetric profile, and it seems the Adobe DNG Profile Editor does the same. If it doesn't it makes something really simplistic, it's easy to see with DNG profiles -- if the DNG LUT (which operates in RGB-HSV space rather than a perceptual space) is very low resolution you can't really do anything serious in terms of tone operator and indeed the output is like that. You can see the compression of red highlights which is a classic property of a simplistic RGB curve etc. I haven't analyzed the general-purpose profiles from basiccolor input, but clearly their focus is reproduction, that's where the commercial interest is. I did make a quicky profile with it and the blues got awfully dark, which is a sign that they just do it the reproduction way with a curve slapped on top.
Here's a few things a general purpose profile needs to consider, that a reproduction profile needs not:
- apply a contrast curve of some suitable shape
- deal with the color appearance changes caused by the contrast curve, largest aspect saturation but also other more subtle things
- gamut compression
- rendering of partially clipped colors
- transition into whitepoint
- robustness
Of the profile makers you mention I don't think anyone really consider any of those aspects. The above list contains much subjectivity, and on top of this you can and further subjective edits, some purely subjective, some in the borderland between subjectiveness and perceptual simulation. Warming up midtones and highlights in yellow-green range while cooling down shadows is one such thing in the borderland, a popular approach in many of the manufacturer's profiles, it can make sunlit landscape look more realistic than otherwise due to the mixed light situation (more spectral red content in the illumination of the sun-lit areas than in the sky-lit shadows)
Skintone appearance depends
a lot on the contrast curve, and how the transition into the white point is made, especially for high key portraits. Reproduction profiles don't need to consider that at all.
The thing is that a standard RGB curve makes sort of a good job in many of those aspects, and it's computationally inexpensive, so it has a strong tradition from back in the days and onward. Adobe made a version of the RGB curve which works the same but holds the HSV-Hue fast, avoiding the typical red-orange hue distortion, but instead causing overly desaturated blue skies for example, and the strong tonal compression of for example red highlights is still there. I haven't analyzed all profile makers on the market, but it seems like using this curve on top of a reproduction profile is the standard approach. The really well-made profiles go past that though.
So even if there would be only one sensor with one CFA (note: there are documented differences in hardware color response even when same sensor is used) the profiles would look very different based on how the mentioned aspects are handled.
Here's an example showing the same camera, with two different profile designs both very well designed but with very different approaches, Leaf's vs Phase One's
http://digitaltransitions.com/leaf-look-iq1-100mp-iq3-100mp/ the largest difference is indeed due to the different contrast curve -- and that is the key element to design when you make a profile (which many profile makers have limited possibilities to actually make precision edits on) -- but the more you look at it the more subtle differences you spot. I prefer the Leaf look of these two.
The more colorful subjects you shoot, the more the profile differences stand out. Vivid flowers for example really push the limits of a profile, pushing it into gamut compression and clipping range. For example, Phase One profiles can be seen to make very good highlight tonality renditions there, perhaps most clearly in red flowers, where many others' idea of just slapping on a RGB(-like) curve compresses and effectively kills tonality. In DCamProf I've spent much time focusing on these types of issues, so I do think it provides something new in terms of what you as a photographer can do. The new stuff is not about target matching, but it's about tone reproduction and all those subjective non-mathematical aspects of profiles.
A general-purpose profile doesn't need a big target like a Colorchecker DC, it will provide marginal improvement over a 24 patch color checker. Big targets are mainly for reproduction work. Why? First a general-purpose profile color correction should be exposure-independent, so it doesn't make sense to profile both a dark red and light red with the same chromaticity. Otherwise with 3D colorimetric correction you need to shoot the same exposure each frame. Making a 2.5D correction a large part of the provided large target patches doesn't make much contribution (just for averaging). Then a general-purpose profile is more sensitive to disturbances in gradients than a reproduction profile, which means that you cannot make strong non-linear corrections, which means it doesn't make sense to precisely correct closely spaced (DE-wise) patches. Large targets are often glossy and very hard to shoot right without significant glare issues. The software can compensate to some extent, but in a mediocre setup the large targets make more harm than good. Thus I don't think that in BasICColor would make a significantly better general-purpose profile with a larger target, than there's something suspicious going on. If it is hard-wired to always make a reproduction-style 3D LUT for correction it might require a large target to just cover the whole gamut so that could be a reason, but if designed to handle smaller target that should not be a problem for general-purpose profile making.
The "robustness" point mentioned above is sort of generic, it's about that you must be really careful about non-linearities, be careful so that your corrections doesn't increase noise, and behave well for crazy colors like artifical narrow band lights, and behave well when the exposure is less than optimal and have many clipped elements. A reproduction profile is used in controlled conditions, clipping shouldn't occur at all, you shouldn't dig into shadows more than necessary etc, so it doesn't need to be robust in the same way.
So how does all this tie back to GFX vs X1D color discussion? I'd say don't worry so much about the hardware, but really look at the end result you get with the raw conversion pipelines provided. The native conversion should be the reference, but try also more popular raw converters like Lightroom and see what the profiles there can do. And of course you could/should try making your own profiles, but making a good general-purpose profile is not as easy as one may think with the currently available software.
