You're up, MP.
[I wrote this quickly, so it's sloppy, probably not that well structure, and full of typos I would assume]
There's a line, it came from Mencken, that went like this: "For every complex problem there is an answer that is clear, simple, and wrong."
I always think of that one when discussions of Keirsey's temperament system arise.
Often, the proponents of Keirsey's work say it is easy to understand, and easy to apply quickly, and really makes the world nice and tidy. Perhaps so, but it does these things by making totally invalid inferences.
What are the problems with the temperament model?
Well, as you noted, Keirsey threw out the cognitive processes. What he is left with is a system where you can only categorize people based on their behavior. In a nutshell, it's a psychology measure that never questions how anyone thinks.
That obviously seems silly, but to elaborate, the problem is that what someone does, like how punctual they are or how cleanly they are, as well as what sort of tenets people subscribe to, like political ideology, are all things that can be arrived at through many different trains of thought. It means that merely observing that a person has certain habits and certain ideologies does not tell you how they perceived or reasoned their way to that point. Trying to type someone on these observable traits will quickly show people to be typological contradictions in Keirsey's system.
Now granted, you cannot know anything about a person by means other than observable traits, technically. But the cognitive model tries harder to get at someone's abstract thinking (often through words as much or more than deeds), and this also why questions on a cognitive test tend to be very general. Kerisey's approach is ridiculously specific. It actually suggests typing people on observations like whether or not they keep surfaces clear and clean. But that doesn't give you any understand of the person's mind. It basically only tells you what you already just observed.
And let's look at the reasoning in Keirsey's own writing. He has very little logical framework in his books. Much of what he wrote is based on his personal experience. That's inconistent, incomplete, and hard to apply. Personal experience alone is not very reliable. Furthermore, outside of that, he makes the odd choice to reach back to the antiquated temperament systems of people like Hippocrates. His reasoning (which strikes me as incredibly ISTJ) is that this is the pattern that has been passed down for ages, and he basically just figures from there that it has enough merit to imitated. I found this decision to also be poorly reasoned.
And there's the fact that, because of how his sytem works, the temperaments are very archetypical. I don't believe the cognitive types, if properly applied, pidgeonhole's anyone or leaves anybody unrepresented (though if you see the way SimulatedWorl applies it, it does in that case). Keirsey's temperaments do, on the other hand. They are inflexible archetypes which go into too much detail per type profile without accounting from all the range of possibilities. He's typing people on so many behavioral details that it should lead to far more than 4 binary variables, and that in turn should lead to far more than 16 types. But again, he's more content to just smash everyone into those details
And that being said, there aren't even really 16 types, because Keirsey gets so carried away with his temperament idea that it starts to devour the types. Some more than others. Perhaps because he fancies himself an NT, he gave the most comprehensive treatment to the NTs. The SJs he treats the worst. Each temperament follows a pattern akin to just one of its types. The Guardians are ESTJs, the Artisans are ESTPs, the NFs and NTs are more even, but they are basically INFJs and INTPs.
The types that are most different from the archetypical type of the temperament are therefore the most poorly accounted. ISFJs, ISFPs, ENFPs, and ENTJs are sort of poorly explained and represented.
So I stick with the cognitive processes. They are an ecompassing and consistent logical framework which can be worked out deductively and applied flexibly, and they actually reflect a person's thought processes, instead of making hasy conclusions from someone's lifestyle habits.