As KCd explains in a comment, the proof of the PNT in Hardy's time seemed to be intimately connected to the complex analytic theory of the $\zeta$-function. In fact, it was known to
be equivalent to the statement that $\zeta(s)$ has no zeroes on the half-plane $\Re s \geq 1$.
Although this equivalence may seem strange to someone unfamiliar with the subject, it is in
fact more or less straightforward.
In other words, the equivalence of PNT and the zero-freeness of $\zeta(s)$ in the region
$\Re s \geq 1$ does not lie as deep as the fact that these results are actually true.
The possibility that one could then prove PNT in a direct way, avoiding complex analysis, seemed unlikely to Hardy, since then one would also be giving a proof, avoiding complex
analysis, of the fact that the complex analytic function $\zeta(s)$ has no zero in the region $\Re s \geq 1$, which would be a peculiar state of affairs.
What added to
the air of mystery surrounding the idea of an elementary proof was the possibility of accessing the Riemann hypothesis this way. After all, if one could prove in an elementary way that
$\zeta(s)$ was zero free when $\Re s \geq 1$, perhaps the insights gained that way
might lead to a proof that $\zeta(s)$ is zero free in the region $\Re s > 1/2$ (i.e.
the Riemann hypothesis), a statement which had resisted (and continues to resist)
attack from the complex analytic perspective.
In fact, when the elementary proof of PNT was finally found, it didn't have the ramifications that Hardy anticipated (as KCd pointed out in his comment).
For a modern perspective on the elementary proof, and a comparison with the complex analytic proof, I strongly recommend Terry Tao's exposition of the prime number theorem. In this exposition,
Tao is not really concerned with elementary vs. complex analytic techniques as such, but rather with
explaining what the mathematical content of the two arguments is, in a way which makes
it easy to compare them. Studying Tao's article should help you develop a deeper sense of the mathematical content of the two arguments, and their similarities and differences.
As Tao's note explains, both arguments involve a kind of "harmonic analysis" of the primes,
but the elementary proof works primarily in "physical space" (i.e. one works directly with the prime counting function and its relatives), while the complex analytic proof works much more in "Fourier space" (i.e. one works much more with the Fourier transforms of the prime counting function and its relatives).
My understanding (derived in part from Tao's note) is that Bombieri's sieve is (at least in part) an outgrowth of the elementary proof, and it is in sieve methods that one can look to find modern versions of the type of arguments that appear in the elementary proof. (As one example, see Kevin Ford's paper On Bombieri's asymptotic sieve, which in its first two pages includes a
discussion of the relationship between certain sieving problems and the elementary proof.)
But I should note that modern analytic number theorists don't pursue sieve methods out of some desire for having "elementary" proofs. Rather, some results can be proved by $\zeta$- or $L$-function methods, and others by sieving methods; each has their strengths and weaknesses. They can be combined, or played off, one against the other. (The
Bombieri--Vinogradov theorem is an example
of a result proved by sieve methods which, as far as I understand, is stronger than what
could be proved by current $L$-function methods; indeed, it an averaged form of the
Generalized Riemann Hypothesis.)
To see how this mixing of methods is possible, I again recommend Tao's note. Looking at it should give you a sense of how, in modern number theory, the methods of the two proofs of PNT (elementary and complex analytic) are not living in different, unrelated worlds, but are just two different, but related, methods for approaching the "harmonic analysis of the primes".
