170 degrees is perfectly fine.
To talk bacteria, you have to take into account two factors: temperature, and time. Anything between 40 and 140 is good for them, anything above 140 is bad. At the same time, food in the dangerzone that is eaten/cooked/frozen promptly, is fine, because it takes time to build up a colony of harmful proportions.
In this case your temperature is 170 degrees, which is hot enough to kill most common bacteria instantly (Milk is pasteurized in the 160s), and then you keep it at that temperature for hours? The toughest common bacteria (C. botulinum) dies very quickly at 180 degrees...It's the only one I know off the top of my head that doesn't immediately die in the 170 range. And keeping your stock at 170 for 10 minutes or so will kill any C. botulinum that may have somehow found it's way inside it.
You've got nothing to worry about.
(Basics of TDT calculation)
It's down to what the chicken ate while it was alive. Saturated fat sets, olive oil sets if you chill it, but not otherwise, and a number of seed oils do not set (rapeseed for instance). When you make a stock which has solidified fat on top, that's saturated fat, so I'd hazard a guess that the stock where the fat doesn't set means a healthier eating chicken, because it contained less saturated fat to start with.
UPDATE: Thank you to the person who bothered to do the research and said my answer 'might have some merit'. Chickens are no different from human beings - the fats you put in are the fats floating round your bloodstream and depositing in various places; think about corn fed chickens, where the fat composition is slightly different, not to mention the colour of the flesh itself. That will be a partial explanation; when taken together with the fact that not all chickens, even in the same flock or brood, get to eat the same diet, because the pecking order dictates that some free range birds don't always get the pick of the food, explains differences in chickens from the same supplier. Of course, if you can come up with another explanation, I'd be delighted to hear it...
UPDATE 2: Perhaps I should have been clearer. I am not for a moment suggesting that the fats eaten are deposited in their original form, but if you know anything about biology (chickens or otherwise) then you'll know that certain synergies occur, depending what's put in, which change the composition of any fats deposited within the body system. Hence the connection between eating lots of saturated fat and having high cholesterol in humans, for example.
UPDATE 3: Rumtscho: Can't find any scientific evidence so far to prove this theory regarding chickens, but, for interest's sake, and to prove how much of a difference it can make, farmed salmon in Britain no longer has a balanced omega 3/6/9 ratio, as it should do, and still does in the wild. It's because the feed had to be changed, and the consequence of that has been a much higher level of omega 6 in particular. I'm still looking for something on chicken.
UPDATE 4: Now I've had time to look properly, it's not at all difficult to find scientific evidence, there's plenty of it. There's a study carried out by The American Society for Nutritional Science in 2000 comparing the fat deposition (and other metabolic processes) between chickens fed the same diet, but one lot with saturated fat included in the form of tallow, and the other lot with polyunsaturated fats. The fat deposition in the birds fed tallow was greater, and the composition of the fat contained more saturates compared to the polyunsaturated group. These results reflect previous studies (Sanz et al 1999 and 2000).
Effectively, it's like everything else - you get back what you put in.
Best Answer
You definitely can't just leave it on the stove; that'd mean far longer than 2 hours in the danger zone. (See for example How do I know if food left at room temperature is still safe to eat?)
If cooking until it's done is out of the question, you need to try to chill it. Putting the whole pot in the fridge might not be the best approach, though; it could take quite a while to cool down. Even if that's less than 2 hours, you're going to go through another cycle later, and the time is cumulative.
The best thing you can do is probably to use an ice bath to help chill it faster.