Writers are usually urged to only include that which is important to a story. The handedness of a character tends to be unimportant... except when it is important, which is when authors state it, and it tends to be important because it is outside of the statistical mode.
You are speaking of relative importance--importance as perceived by the observer. On those terms, I would disagree with you entirely--which matters not a whit. However, it is hand-specificity's importance in absolute terms that I was getting at previously. Note that I wasn't referring to
handedness per se; I was talking about the much simpler information of which hand a person is using to perform a given task. That information is significant very often, objectively, because the hands are the center of most human activity and usually pertain to any human action happening in a story. Most physical interaction with another person, or humanoid, requires knowing which hand they're using for a given action. More fundamentally, our understanding of our own position and motion requires knowing which hand
we're using (for a given action). In real life we process this information without even thinking about it, taking it totally for granted, but imagine how much trouble we would be in if we didn't have it!
Now, there is a more general, tangentially related point: You mentioned the principle of relevance...of clean, crisp prose not cluttered by extraneous information. Most of the time, hand specification requires only a single four- or five-letter adjective in front of a noun that's already there. The rest of the time, it's a matter of adding one short phrase. From that small investment, a considerable gain can be realized. If the writer specifies which hand their character is using to hold a cup, the reader can and usually will put that information to immediate use. What's really interesting, though, and directly central to this little vignette, is that, by including this information, the reader may be able to see even more than what they are told, purely by association. Try it yourself:
He was holding a cup of tea in his hand.
He was holding a cup of tea in his left hand.
Functionally, these two sentences are identical. But they have two different impacts on the reader: The former is straightforward. Reading it is a passive experience. We understand that the character is holding a cup of tea. But the second sentence provokes our thoughts. In addition to providing us with the basic information of the cup in the left hand, very obviously the mention of the left hand raises the question of what the character's right hand is doing. Less obviously, the asymmetry of the description generates further details in the image.
In visualization, the brain takes all kinds of shortcuts: In a non-visual medium, like a book, have you ever known what a character's voice sounds like or face looks like, even without explicit description by the author? If so, have you noticed that, when you try and pin it down, you don't actually have all the details? The brain uses symbols and associations and partial constructions to illustrate only what it considers relevant; it never paints the entire picture into a literal image as we might see it in reality.
Owing to our evolutionary past, lateral information about animals and especially fellow humans is one of these things that people consistently notice. In fact, the impressiveness of how easily artists neglect to include such information in their writing and drawing is more likely a testament to the significance than insignificance of this information: Specifically, I propose it is so important that it predates our higher consciousness. Thus, while many people don't think to include it in their work, everyone reacts to it--whether they realize it or not. Asymmetry in the human form is very significant to us. You don't even need to take my word for it; you probably already know it. Anything that can appeal to our older brain structures is significant to the skillful writer, who can exploit our biases to describe scenes without stretching into the uncontroversially disadvantageous realm of verbosity and superfluity. (Yeah, yeah, I know....) Thus, the mere mention of a word like "left" or "right" adds a hefty spatial dimension into the author's description, which the reader's brain readily seizes upon to create even more imagery. Try it yourself with other sentence constructions that are hand-neutral and then hand-specific.
Lastly, although I am not implying that you would fail to make the distinction, nevertheless I want to point out the distinction between the quality of a piece of writing and its level of detail. They are independent variables. It's certainly possible for a writer to incorporate hand-specificity in a way that detracts from the quality of their work, but this is not inherently because they are being verbose or straying off-topic. Any such detraction is more likely due to deficiencies in the writer's skill or technique.
