Is there a simple way to prove Bertrand's postulate from the prime number theorem?
The prime number theorem immediately implies Bertrand's postulate for sufficiently large $n$, but it fails to establish a base case (the linked proof on Wikipedia explicitly gives the base case $n \ge 468$). In the other direction, Bertrand's postulate yields $\pi(n) \ge \log_2(n)$ which seemingly adds little beyond Euclid's theorem to any proof of PNT. Is the prime number theorem really "stronger" than Bertrand's postulate, in the sense that assuming the former can simplify a proof of the latter? What lemmas are needed in addition to PNT for a more concise proof of Bertrand's postulate?
EDIT: I am specifically referring to this version of PNT:
$\pi(x) \sim \frac{x}{\log x}$