Word or phrase for the phenomenon (or thinking error) where people will be satisfied with an answer simply because it was difficult to arrive at?
As per the title, I'm looking for a word or short phrase that describes the phenomenon where people will accept an answer or solution that was difficult to arrive at, simply because it was difficult to arrive at. For example, a police detective making an investigation, coming up with a suspect, but not vetting that suspect because they're confident they've done enough in the first place; the concept is more general than that example though.
My thoughts so far are 'cognitive overload' or 'sunk-cost fallacy', but they don't quite hit the nail on the head. I suppose 'laziness' also kind of describes it, but I'm looking for something more specific.
Cheers.
Edit: As requested, here are some other examples:
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A student is working on a difficult math equation. The student apparently solves the equation after a moderate amount of time and effort. However, the student does not bother to check their working or the potential for other answers, even though he is not pressed for time, tired, or has any other pressing issues, but simply because of the phenomenon I'm looking for.
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A thief is looking for something precious in a home they have broken into. The resident of the home has a safe that is relatively difficult to find and break into and has some nominally valuable stuff in it. However, this is a red herring for another safe that is better hidden and has more valuable stuff in it. The thief is satisfied with the red herring safe and leaves without checking for anything else of value, even though he's not concerned about being discovered, but simply because of the phenomenon I'm looking for.
Edit 2: edited the title to note that it doesn't necessarily have to be a thinking error, but that is still the direction I'm heading towards: ie. it is a problematic phenomenon, or at least the phrase I'm looking for has that connotation.
Solution 1:
Cognitive Psychologist Daniel Kahneman addressed this type of cognitive shortcut in his book, Thinking, Fast and Slow. In it, he hypothesizes about two mental systems at work.
"System 1" is fast, instinctive and emotional; "System 2" is slower, more deliberative, and more logical.
Wikipedia at "Thinking, Fast and Slow"
So you are describing a System 1 bias.
This is likely an Article Substitution:
Substitution
Main article: Attribute substitution
System 1 is prone to substituting a simpler question for a difficult one. In what Kahneman terms their "best-known and most controversial" experiment, "the Linda problem," subjects were told about an imaginary Linda, young, single, outspoken, and intelligent, who, as a student, was very concerned with discrimination and social justice. They asked whether it was more probable that Linda is a bank teller or that she is a bank teller and an active feminist. The overwhelming response was that "feminist bank teller" was more likely than "bank teller," violating the laws of probability. (Every feminist bank teller is a bank teller). In this case System 1 substituted the easier question, "Is Linda a feminist?", neglecting the occupation qualifier. An alternative interpretation is that the subjects added an unstated cultural implicature to the effect that the other answer implied an exclusive or, that Linda was not a feminist.
(op. cit)
In your examples, the detective has substituted an easier question, "Is Mr. X the culprit?" instead of asking the harder question, "Who is the culprit?"