Let's start with the most basic statement: almost every DSLR or EVIL (okay, "mirrorless") camera made in the past couple of years is perfectly adequate to the task. Portraiture isn't exactly demanding of camera or lens as long as you avoid the worst that's out there. Medium format backs will give you slightly better tonality than full-frames, which will in turn give you slightly better tonality than APS-C cameras, and so on down the road. Larger sensors will give you more apparent detail than smaller ones, and sensors without AA (OLP) filters will give you more actual detail than equivalent sensors with them. Portraits, though, don't usually demand (and sometime actually reject) the sort of detail you'd want in a commercial beauty or fashion shot, and minor differences in tonality for shots made under controlled studio conditions are really only visible if you put two shots side-by-side for examination.
You'll want to have a lens or lenses of the appropriate focal length for the pictures you want to make, but what, exactly, constitutes a "portrait lens" depends entirely upon your style, not on something somebody wrote in a book somewhere. If the fixed focal length lenses on the compacts you mentioned aren't right for you, then it doesn't matter at all what the image quality of the system is, the camera won't work for you. If your style depends on a razor-thin depth of field, then you won't be easily satisfied with a small-sensor or APS-C camera; you'll need to look at full-frame 35mm format or larger.
All in all, though, the quality of your images is going to depend an awful lot more on your lighting and environment than on the camera. That doesn't mean that you necessarily have to invest huge chunks of money into studio lighting, but that lighting will be your primary concern, and any camera choice that results in compromised, inadequate lighting is going to have a large negative effect on your portraits. Look after that part first. (And again, what constitutes "adequate lighting", beyond the necessity of getting the ISO down and the shutter speed up, largely depends on your style. It may be a couple of speedlights and some homemade reflectors, it may mean several constant light fixtures, or it may mean a half-dozen monolights or a couple of packs with multiple heads, along with reflectors, softboxes and grids.)
But there is one thing you really need to keep in mind: no matter what your finished product looks like, if you show up with the same camera that your subject's nephew got for his birthday last year (or one that looks too much like it) you're going to have a hard row to hoe. You won't be dealing with fellow pros who understand the choices you're making; you'll be dealing with civilians who have expectations. An SLR makes a statement that a mirrorless does not. The Sigma DPx cameras are a perfect illustration—it doesn't matter that they cost a thousand bucks and make near-MF-quality images at low ISOs, they look like a pocket point-and-shoot (they don't even zoom), so even if there were a hypothethical DP3 Merrill with a superb 60mm/1.4 lens on it¹, it would be the last thing you'd want to use as a portraitist unless your name was already a household word. Even if your shots are truly spectacular, the customer is going to assume that it was a fluke since you don't even own a "pro" camera, and the word-of-mouth is going to have more to do with your equipment than your images. Anybody who tells you different has probably never tried the pro route.
Building a business is a lot like going to the bank to ask for a HUGE loan. You definitely want to "suit up", even if that isn't who you really are. An SLR of any description at least looks to the customer like the right kind of gear. (A huge MF camera is quite beyond their understanding; all they know for sure is that nobody they know has anything nearly that big, so it must be good.) The big names can afford to be quirky with their choices because their reputations precede them; a novice pro can't afford not to look like the stereotype.
¹ Since this was written, the Sigma DP3 Merrill went "unhypothetical", but it has a 50mm lens rather than the 60mm I postulated. I really must fine-tune my precognitive powers some day.