While I can't speak directly for Hardy and Wright, I think the following is a plausible explanation, based on a copy of the fourth edition of H&W. Just prior to claiming the equivalence of $\pi(x) \sim x/\log x$ and $p_n \sim n \log n$, they spell out the argument that the inverse function of $x/\log x$ is asymptotic to $x \log x$. For completeness, here is the brief argument, very mildly paraphrased:
If $y = x / \log x$ then $\log y = \log x - \log \log x$. Since $\log\log x = o(\log x)$ we have $\log y \sim \log x$ and thus $x = y \log x \sim y \log y$.
The point here is that this argument illustrates a "moral": the key observation is that once we establish that $x$ and $y$ are not too far apart (that is $\log x \sim \log y$) then we can justify shifting between them and that this allows us to (asymptotically) invert functions which do not have nice inverses. Imagine now $\pi(x)$ in the place of $y$, not in an exact copy of the above proof, but a modified version with this moral intact:
If $y \sim x / \log x$ then $\log y = \log x - \log \log x + o(1)$. Since $\log\log x = o(\log x)$ and $o(1) = o(\log x)$ we have $\log y \sim \log x$ and thus $x \sim y \log x \sim y \log y$.
This establishes Theorem 8 in a way that strongly echoes the preceding discussion (and without making any general claim of asymptotics of inverse functions). Likewise, the same argument goes through for $\asymp$ instead of $\sim$, with $O(1)$ replacing $o(1)$.
I do agree there is sloppiness in saying that this inference follows "from the remark" and not in a manner akin to the remark. It's also interesting that they take the trouble to write out a proof of Theorem 9 ($p_n \asymp n \log n$) from Theorem 7, but refer to Theorem 8 ($p_n \sim n \log n$) as a trivial consequence of Theorem 6 (the antecedent theorems in both cases being the corresponding estimate on $\pi(x)$). I'm inclined to chalk this up to human fallibility :).