As always, personal opinion.  There are many different ways to humanely slaughter chickens.  To me, the key is that it be quick and sure.  Once the nerves are severed the chicken does not feel anything.  It will have certain reactions and body movement, but these are nervous reflexes, not consciously thought out movements.  
One place I am looking at slaughters on the farm, is uninspected and "labels" the meat "not for human consumption"--I am thinking this may actually be a good thing?? They also allow us to visit so we may go out there.
Why do you think that a place that are taking steps to avoid inspection would be a good thing?  Each State has different regulations, so I don't know what the rules are in your state.  Heck, I don't even know what they are for mine.  I am not a commercial business so I don't have to worry about getting licenses and meeting basic cleanliness rules.  But yes, I think a visit by you would be important.  
However, if they are raising birds and not just slaughtering other people's birds for them, don't be too surprised if they restrict you from looking at certain parts of the operation.  Biosecurity should be important to them.  Some places require people to totally change clothing or at least put on coveralls and wash their shoes in a disinfectant bath.  Delivery trucks may have to roll through an antiseptic bath to disinfect their tires.  If you come visit, I'll show you my chickens without doing any of that, but i don't have thousands of chickens that can be wiped out by someone tracking in a chicken disease on their shoes.
And why not Cornish X? Are they sort of a Frankenbird--like, bred so much for meat so as not to be self-sufficient?
I don't know why someone else was so adament against Cornish X.  I'll not assume to guess what goes in in someone else's brain.  Let them explain where they are coming from. 
The Cornish X, also known as broilers, are a specific breed developed to maximize weight gain per pound of food they eat.  They generally reach slaughter age at about 6 to 8 weeks, so they are extremely tender and don't have the deeper flavor of the dual purpose birds.  Many people think the tenderness and blander flavor has to do with breed.  I think it has a lot more to do with age, somewhat like veal is more tender and blander than beef.  That may not be a great analogy because veal is also mainly milk fed as opposed to pasture or grain, but I really do think age has a whole lot to do with it.  They can be raised organically and they can be pastured.  There are people on this forum that do.  But, no, they are not self-sufficient.  If allowed to eat what they want, they gain so much weight that their body breaks down from growing too fast.  You can keep them and breed them by severely restricting their diet.  That is what the commercial operations pretty much do, although that gets a bit more complicated to explain.  But even if you do restrict their diet, you still often have high mortality.  
If you are really interested, this video helps explain how they breed them.