What is the expected number of children that need to be born for every possible point mutation to occur once?
I don't think the book's reasoning is correct.

There are about 3 billion base pairs in the human genome, but only ca 25.000 protein-coding genes. From the top of my head, I recall that each child has about 30 mutations. If that's in the total genome and if they were all point mutations and there's 4 options for the base pair, then we have $30$ mutations out of 12 billion options in each child. So that each possible point mutation has a probability of $30/12 \text{ billion} = 2.5 \times 10^{-9}$ or $1/(2.5 \times 10^{-9})$ if you like. The book's number is 23 times larger but in the same ballpark range. Lets call this probability $p$. 

Now the probability of a mutation occuring in n children at least once is equal to one minus the probability of it never occuring in n children. In other words:
$$1 - (1 - p)^n =$$
$$1 - (1 - 2.5 \times 10^{-9})^n =$$
$$1 - (0.9999999975)^n$$

If we want the mutation to occur with probability 95%, we have to solve:
$1 - (0.9999999975)^n = 0.9$
This has the solution $n = 1 \times 10^9$

In other words, with 1 billion children, each point mutation has a 95% probability of having occured at least once. Mathematically this is not the same as saying that every mutation will have occured. In fact, this implies that 5% of the mutations will not have occured. But perhaps surprisingly, we only need 4-5 billion to up the chance to 99.999%. That still leaves out thirty thousand point mutations, but is sufficiently close to "everyone". 

Compare this to the numbers you mentioned:
n = 17 million gives each gene a 4% chance of occuring
n = 80 million gives each gene an 18% chance of occurring.

Secondly, a mandatory rant. :-) "The Perfect Health Diet" is written by Paul and Shou-Ching Jaminet, an astrophysicist and molecular biologist respectively. This has the potential to marry the physicist's penchance for oversimplification with the biologist's lack of mathematical prowess. We see both in the passage you ask about. A shoddy calculation about an oversimplification of what mutations and evolution is about. 
Both the book's dustcover use of a word like "detoxifying" and mention of how diseases spontaneously resolve if you eat right, are both strong crackpot indicators. It's the kind of thinking that killed Steve Jobs because he didn't seek medical help soon enough but tried to eat his way out of his illness.