How about sports and fast paced games?
Players are often required to make decisions with no time whatsoever to plan. For example, you might find yourself surrounded by enemies with no warning.
You need to know whether to run on foot, to teleport away, or to fight.
The difference between reacting in a third of a second and a fourth of a second could mean life or death.
Success in this situation, assuming it's possible, is dependent on your experience in similar situations and your instinctual reaction. Since you do not have the time to think, your decision is almost guaranteed to be imperfect, but any improvement in it is highly beneficial.
Obviously, the same would often apply in war or in certain crisis situations.
You mention lots of fields (computers, math, science, engineering) where your argument is almost tautological: in a case where you have time to reconsider each decision, a slow but reliable and precise method is better than a snap judgment. Yes, I would agree with you, and I would also agree that logical thinking is better than intuitive thinking in many, many situations.
Are you suggesting that the ability to model others or respond to nonverbal cues is innate, rather than learned? I would definitely disagree, though proving it would be difficult. I suspect that it's a matter of internalizing the results of numerous actions and reactions in different situations. In my experience, it's often developed by people who travel lots or are otherwise exposed to tons of different people in a situation where being friendly and getting on their good side is very helpful. Some of them, pretty bad at socializing before they were in such a situation (and really gave it the necessary effort to learn).
I disagree, however, when you say that being socially successful is innate.
The advantage of gut feelings and intuition lie with their ability to synthesize years of experience and thousands of variables into one answer within less than a second.
When is this necessary?
During a conversation, someone watching your face is going to be observing how you react (even in the smallest possible ways) as they speak. You don't have an hour, five minutes, or even two seconds to decide how to present yourself; they're going to judge you based on that instantaneous reaction (or a lack of one, including a delayed reaction or straight face.)
Anyone who is a natural "people person" - the kinds of people who can get almost anything they want from anyone around them, who make great salesmen or politicians - is going to need to be able to continuously react "properly", and that means intuitive judgments.
Same thing with any kind of games/sports, or literally any other situation where a quick reaction is required and not immediately responding will doom you no matter what.
A metaphor: Knowledge is a jigsaw puzzle, and the search for truth is a process of trial and error fitting new pieces alongside those already have. The more pieces you have in place, the quicker you can accept or reject new ones; the more granular the detail you perceive in their edges, the better you can identify the exact shape of holes in the puzzle and make new discoveries.
And if there's a misshapen piece you absolutely refuse to move it will screw up the entire puzzle and you'll never get it right. This method is great - generally reliable sources which fit together are free pieces which act as your foundation to even get started.
Unfortunately, it's often easy and natural to force contradicting new data into your existing model even if it really doesn't fit - patching the conflicts without ever really noticing the dissonance, and overfitting your theory without actually restructuring your beliefs. One useful trick for checking yourself: explicitly asking yourself "what do I expect this figure / fact to be or say?" on each step of the project before you look it up. If you go in with reasonably certain expectations and the data reads wildly out of bounds, maybe you've found a major hole in your understanding of the issue, maybe the info is bad, or maybe that figure is saying something very different than you interpreted it.