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.
Given the context of the original statement you want to paraphrase, you should not use either term.
The specific language you want to paraphrase is
I think it can be due to our upbringing. We want what we want when we want it.
You've proposed
today’s teens are much too pampered and as a result, lazy.
You've got a great start with reformulating the sentences so that you can replace "due to" with "as a result". You have also (like the narrator of the original piece) paraphrased the second half of the quote as "lazy", which works fine.
Your sticking point is how to paraphrase "our upbringing". You have proposed either "pampering" or "spoon-feeding" as replacements, but neither is really appropriate here.
When paraphrasing someone else's words, it's very important not to mis-characterize the original meaning. In this case, the original speaker talked about "our upbringing" and its results ("lazy" and "want what we want when we want it"). But she did not specify what aspect of rearing caused these result.
You have inferred that these lazy young people's parents were "pampering" or "spoon-feeding" them, and that has caused them to behave this way. But perhaps, instead, Ms. Benoit meant that she and her peers had been particularly deprived for most of their childhoods, and now that they don't have to work as hard they don't want to. That scenario might not seem as likely to you (it doesn't to me, either) but the original statement leaves open the possibility.
So, what should you use?
Look for a neutral synonym for "upbringing", or just use "upbringing", and then leave the specific deficits of this to the imagination of the reader (like the original speaker did). You may need to rearrange your sentence slightly to make this work. I would suggest something like
today's teens are lazy as a result of their upbringing.
or
as a result of the way today's teens were raised, they are lazy.
or, to stick as close as possible to your original structure,
today's teens were raised to be entitled and as a result, lazy.
Note that in that last example, I have substituted the adjective entitled for the statement that teens "want what [they] want when [they] want it", as suggested by Rhythmatic. That works, since you actually do have two descriptions of the young people in the original, and you were previously using the single word "lazy" to stand in for both. What we don't have is a description of the teens' parents' parenting style.
Best Answer
I think the problem you're having is that you're mixing two sentence types - affirmative and negative sentences. The first sentence is an affirmative sentence about a negative thing... You're saying he is something, but the thing you're saying he is, is a negative thing.
The next sentence says he isn't something, so therefor it has to follow the rule for negative sentences by using the word "either."
If instead, the next sentence said he was something, rather than he wasn't something, then you could use the word too.
Examples:
He is lazy. He is stupid too.
He isn't lazy. He isn't stupid either.
He isn't lazy. He is smart too.