You have opened the proverbial can of worms Colin. Periodically the issue of Pyro developers is discussed by the smart and very talented people on this site. As you will shortly read, there are as many different opinions as there are participants. Some swear that Pyro offers benefits that are unique and easily seen. Others say the opposite: Pyro offers little in the way of recognizable differences in prints that are made from their negatives developed in their "standard" developers. Some of the statements regarding Pyro can be verified ( see Ed Buffaloe's site and others ) using sophisticated densitometers, while other claims cannot be verified ( see the more easily understood work of Howard Bond and the late Phil Davis ). Frankly, I don't know if there is a "correct answer" about the advantages of Pyro except to say that the appearance of negatives developed in Pyro ( the stain and the "detail" ) excites those who use one of the various formulae. Indeed, to read the statements of Gordon Hutchings makes one want to run out and buy some pyro simply to experience the rush of excitement that Mr. Hutchings claims to get when he sees one of his "Pyro negatives"! However, to be fair, to look at Mr. Hutchings work is certainly to be impressed with what he has accomplished using Pyro. Gordon is rightly held in very high esteem indeed, and is one of those who is no doubt responsible for the renaissance of Pryo developers. However, when one looks at the work of photograpers who eschew Pyro for various reasons ( one having to do with the fear of deleterious health effects from Pyro ) one sees work that is equally well done. In my comments, I don't refer only to conventional silver prints made by enlarging, for one can easily use conventional developers to develop negatives to a density that can be used in contact printing using some of the alternate techniques. A caveat: I am not experienced in Platinum, Palladium, Azo, etc. printing and so I cannot render a meaningful personal statement based upon my own work concerning Pyro negatives using these and other alternate techniques. However, I can tell you that I have made 8x10 contact prints from negatives developed in Pyro and DDX on the same day using fresh Dektol and fresh paper and the same light source. I have endeavored to make the prints to the same Dmax and the same contrast, but have not used a densitometer to be certain that I was successful. I am certain that there were small differences in the prints. However, if I put the prints away, and pick them up "blindly" in a month or so, I am not certain that I can tell the contact prints made from Pyro negatives from those made with my conventional developers. I am sure, however, that there will be those with much more talent and experience than I who will tell you that contact prints made from Pyro negatives are infinitely better for a variety of reasons, and that the reason I see no difference with Pyro has to do with my developing time and technique, or my printing set up. I admit that my lack of experience makes me unable to completely refute their assertions. I will go back and try again, for Pyro must be better....;}, but I am just not capable enough to bring out the differences. Of course, such opinons might well be correct...I think....
And so, there will be no "correct answer" to your interesting and relevant question. There will only be, in the end, personal opinions from many very well meaning and accomplished analog photographers. What you might consider doing is trying to develop a few of your negatives in the Pyro developer of your choice, and then print the negative along with the same scene developed in your standard developer. As Phil Davis has stated, any meaningful comparison depends upon the difficult chore of making each print to the same DMax on the same paper toned the same way, etc., etc.. Of course, frame the prints in exactly the same exact way, and then have impartial "blinded" ( in the statistical sense ) observers look at the prints in random order in the same viewing light. Only then will you be able to say if the difference is/differences are significant as reflected in your prints. You will then become either an accolyte, or a skeptic! However, trying is half the fun...go to it, and let us know what you find out!
Stay well.
Ed