A lot of the time, new legislation is not intended to create new laws, but to make changes to (amend) existing ones.
This is useful because, when looking at the new legislation, it makes it clear precisely what is being changed. Parts of the old law that are not being changed don't need to be mentioned in the new one, and so don't need to be debated or voted on.
The downside to this, as the question notes, is that the new legislation is difficult to read, and makes no sense unless you also have access to the law being amended.
It's not just US legislation that is like this! It's common in British law too. For example, every year, the government changes the taxes on alcoholic drinks. The taxes were created by a law passed in 1979. They only want to change the numbers, nothing else, so section 186 of this year's Finance Act looks like this:
186 Rates of alcoholic liquor duties
(1)ALDA 1979 is amended as follows.
(2)In section 5 (rate of duty on spirits), for “£25.52” substitute “£26.81”.
(3)In section 36(1AA) (rates of general beer duty)—
(a)in paragraph (za) (rate of duty on lower strength beer), for “£9.29” substitute “£9.76”, and
(b)in paragraph (a) (standard rate of duty on beer), for “£18.57” substitute “£19.51”.
(4)In section 37(4) (rate of high strength beer duty), for “£4.64” substitute “£4.88”.
(5)In section 62(1A) (rates of duty on cider)—
(a)in paragraph (a) (rate of duty per hectolitre on sparkling cider of a strength exceeding 5.5 per cent), for “£233.55” substitute “£245.32”,
(b)in paragraph (b) (rate of duty per hectolitre on cider of a strength exceeding 7.5 per cent which is not sparkling cider), for “£53.84” substitute “£56.55”, and
(c)in paragraph (c) (rate of duty per hectolitre in any other case), for “£35.87” substitute “£37.68”.
It doesn't make for exciting reading, but it avoids having to rewrite whole sections of the old law in the new one.