Why it’s wrong
Yes, it’s grammatically wrong. But your way of thinking about English grammar is right in a very profound way: wondering if singular and plural could be exploited to suggest some subtle distinction.
Here’s why it’s wrong. The pronoun their calls for a plural antecedent. Hair is singular. Also, movement is singular, so it has to be grammatically tied to an individual hair. If you tie it to the mass of hairs all together, then it suggests that the mass of hairs all together (what we normally call “hair” with no determiner) has one movement as a whole.
How to do it right
There are ways to use singular vs. plural to indicate what you have in mind. Here’s how you’d do it:
… each of his brown wavy hairs had a movement of its own…
The singular word each gives the singular movement something to attach to, and hairs is plural. This makes it clear that the sentence is talking about many movements, not just one.
You could also indicate separate movement of each strand of hair like this:
…his brown wavy hairs had movements of their own…
You are right that logic often trumps rigid grammar rules, leading a reader to interpret a sentence reasonably when too-strict application of grammatical regularity would lead to clumsiness. But since the language provides a straightforward way to indicate the intended meaning in this case, there’s no pressure to bend grammar.
The inevitable complexity
Their can take a singular antecedent when it stands for a person and you’re trying to avoid indicating “their” gender. However, to many people's ears, this usage sounds sloppy or ungrammatical, or at best informal, because their calls for a plural antecedent. There is currently something of a war going on in the language right now, to allow their to refer to a singular person as antecedent in order to avoid sexist language. Perhaps after that war is won, their will broaden to allow singular antecedents of all kinds, but today such a development is beyond the horizon.
The rule says to use the plural to mean either "all" or "any" of that thing. You have to figure out which of these from the context.
Someone ate the donuts (they ate all the donuts that were in a particular place)
I bought us donuts (I bought some quantity of donuts)
She hates donuts (She does not like (to eat) donuts of any kind)
Basic logic applies here. If a man has three apples and you say "The man is eating apples" it would be silly to assume he's eating all the apples in the world. The more reasonable assumption is that he will eat at most those three apples.
Of course he could eat one, or two, or all three of the apples. The exact number is unspecified, and unimportant. The point is that he is eating apples, and not something else.
Outside the window, I see birds flying. (It doesn't matter how many birds, just that there are birds)
My mother bought me shoes for my birthday (It could be one pair, or multiple pairs of shoes. Which is not important, the point is that I was given shoes)
This room is where they store computers before they sent to the customer (The number of computers is not important, only that this room is used for their storage).
Best Answer
When we say "I go to work by car", we are referring to 'car' as a mode of transport, rather than a subset of vehicles. As a single mode of transport, we use the singular form of the word car.
On a related note, modes of transport (by bus, by train, by car, by plane, by bicycle) don’t have an article. Which is why we don't say "I go to work by the car."