This is sort of a two part question, why is transporting legal US tender illegal, or if not, require some sort of approval? Secondly, who sets the number at $10k? I remember that being the number as a kid 20 years ago from movies, but $10k 20 years ago is worth $16k now... it's not a terribly large amount of money any more.
Where did we get such a strange restriction?
It doesn't even have to be that much. Carrying a lot of cash gives police cause, under the current laws, to confiscate the money without even charging the person with a crime.
Never mind if you are a person who doesn't believe in banks or just won the cash at a casino. Police don't seem to care about any explanations as to why a person has cash. They just take it and it's up to the individual to hire a lawyer and try to get some of it back.
Banks have to report it if people deposit $10,000 or more. Then a new law allowed police to confiscate a person's bank account if they deposited less than $10,000 because they can claim the person is trying to get around the law by depositing less than $10,000. In other words, they want your money and they will take it. Some business owners have had their accounts seized after they made deposits of less than $10,000. One person ran a cash only business selling food. Perfectly legit, but lost everything anyway.
In many of these cases, the people can easily account for the money and there is nothing illegal. Of course, when government decides to make it a crime just to have the money, regardless of how legitimate it is, it allows police to take it without even accusing anyone of wrong doing, let alone actually charging them.
What this law does is give police authority to take anything from anyone for no reason at all.