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nickernst comments on The Trolley Problem in popular culture: Torchwood Series 3 - Less Wrong
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Utilitarian calculation is a more rational process of arriving at a decision, while for the output of this process (a decision) for a specific question you can argue that it's inferior to the output of some other process, such as free-running deliberation or random guessing. When you are comparing the decisions of sacrifice of children and war to the death, first isn't "intrinsically utilitarian", and the second isn't "intrinsically emotional".
Which of the decision is (actually) the better one depends on the preferences of one who decides, and preferences are not necessarily reflected well in actions and choices. It's instrumentally irrational for the agent to choose poorly according to its preferences. Systematic processes for decision-making allow agents to explicitly encode their preferences, and thus avoid some of the mistakes made with ad-hoc decision-making. Such systematic processes may be constructed in preference-independent fashion, and then given preferences as parameters.
Utilitarian calculation is a systematic process for computing a decision in situations that are expected to break intuitive decision-making. The output of a utilitarian calculation is expected to be better than an intuitive decision, but there are situations when utilitarian calculation goes wrong. For example, the extent to which you value things could be specified incorrectly, or a transformation that computes how much you value N things based on how much you value one thing may be wrong. In other cases, the problem could be reduced to a calculation incorrectly, losing important context.
However, whatever the right decision is, there normally should be a way to fix the parameters of utilitarian calculation so that it outputs the right decision. For example, if the right decision in the topic problem is actually war to the death, there should be a way to more formally understand the situation so that the utilitarian calculation outputs "war to the death" as the right decision.
I'm not convinced utilitarian reasoning can always be applied to situations where two preferences come into conflict: Calculating "secondary" uncertain factors which could influence the value of each decision ruins the possibility of exactness. Even in the trolley problem, in all its simplicity, each decision has repercussions whose values have some uncertainty. Thus a decision doesn't always have a strict value, but a probable value distribution! We make a trolley decision by 1) Considering only so many iterations in trying to get a value distribution, and 2) seeing if there is a satisfying lack overlap between the two. When the two distributions overlap too much (and you know that they are approximate, due to the intractability of getting a perfect distribution), it's really a wild guess to say one decision is best.
Utilitarian calculation helps the process, by providing means of deciding when each value probability distribution is sharply enough defined, and whether the overlap meets your internal maximum overlap criteria (presuming that's sharply defined!), but no amount of reasoning can solve every moral dilemma a person might face.