An otherwise 'brilliant' person can be slow at solving certain problems, yes. And no, cognitive performance is not the same thing as running a marathon. A marathon measures your ability to reach a certain point in a certain amount of time. An intelligence test (or any academic test, for that matter) assesses to make sure that you have learned the material well enough that you can correctly answer a subset of questions pertaining to the material within a certain time constraint. The motivations here are completely different.
That being said, I would argue that the ability to answer questions rapidly has less to do with intelligence and more to do with executive functioning. For the sake of argument, I am going to use the word 'genius' to describe what you may otherwise be describing as a 'brilliant' person:
A genius is a person who displays exceptionally superior intellectual ability, creativity, or originality, typically to a degree that is associated with the achievement of new advances in a domain of knowledge...There is no scientifically precise definition of genius, and the question of whether the notion itself has any real meaning has long been a subject of debate, although psychologists are converging on a definition that emphasizes creativity and eminent achievement.
Notice how this definition capitalizes on creativity, originality, and an understanding of a domain of knowledge to the point where he or she can advance the field forward. Notice how there is no mention of cognitive speed or processing speed outlined here. However, take a look at the definition of executive functioning:
Executive functions (also known as cognitive control and supervisory attentional system) is an umbrella term for the management (regulation, control) of cognitive processes, including working memory, reasoning, task flexibility, and problem solving as well as planning and execution.
Sounds like executive functioning plays a large role in one's ability to rapidly solve problems. And this makes sense.
Consider "wonder drugs" (amphetamines, Modafinil, even caffeine) that typically boost cognitive performance and processing speed. One can take a drug to become more focused, and to increase their executive functioning. Amphetamines in particular (such as Vyvanse, Adderall, Ritalin) are drugs that are commonly administered to children and adults who struggle with ADHD -- a disorder caused by "significant problems with executive functions." 1 These drugs work by increasing norepinephrine and dopamine in the brain, thus temporarily boosting cognitive performance. Would you call a medicated adult with (ordinarily) poor executive functioning 'brilliant', as opposed to the unmedicated adult? To do so wouldn't make sense, unless we are to assume that one's intelligence is malleable based on the drugs that they take -- in which case, perhaps IQ tests (and achievement tests in general) are worthless.
I would reject the definition for 'intelligence' given earlier
The ... capacity of the individual ... to deal effectively with his environment
because it can be used to refer to a multitude of things. The 'capacity of the individual to deal effectively with his environment' would more easily fit the stereotype of the salesman than that of the absentminded college professor. Take this description of Albert Einstein:
[Einstein] was forgetful, could never find his keys and often seemed oblivious to his surroundings.
And take this description of Isaac Newton, taken from the Oxford Press:
Quarrelsome and quirky, a disheveled recluse who ate little, slept less, and yet had an iron constitution, Isaac Newton rose from a virtually illiterate family to become one of the towering intellects of science.
None of these descriptions leave me with the impression that these otherwise-intelligent men had a knack for 'deal[ing] effectively with his environment'. Is the environment not external -- the antithesis of mental?