It's definitely late in the game to this question, but Martin kindly pointed me here, and I think there's something else to be said.
The search for the elementary proof has two points of view I think are related to the soul of mathematics as it stood during the time when the preoccupation was great.
Further Advancement of the subject through new techniques.
The idea of an elementary proof meant that we would necessarily need to come up with new ideas that we had not before in order to produce the proof that would prior only yield to techniques using analytic methods. It is very common in mathematics to have several proofs of the same results. Oftentimes new proofs either simplify old ones--allowing easier transmission of the ideas and oftentimes bringing in new viewpoints which allow the theory to take a large step forward.
Tate's thesis allows one to prove the functional equation for the $\zeta$ function. This is something we've known for some time, but the ideas present in the new proof, allowed for the roots of incredibly important new mathematics to be developed because of how he did it. One can see very slick, elucidating proofs which somehow seem to be the "right" ones. Erdös would probably say they were "proofs from the book" which is the namesake and inspiring spirt of this book. The proof of the PNT equivalence with the non-vanishing statement mentioned in the previous answer comes from a proof of the PNT through the Wiener-Ikehara theorem, another idea that has more widespread applications than just the purpose to which it is put with the PNT.
And this is not a phenomenon unique to number theory. We have a proof of the classification of surfaces via Ricci flow, decades after the original proof, which motivated the idea that techniques using Ricci flow might give the classification of $3$ manifolds (Perelman's celebrated proof of the Poincaré conjecture proved this and more to be true.) Many old proofs in modular arithmetic are much easier to prove using the language of groups, such as the fact that $\left(\Bbb Z/p\Bbb Z\right)^*$ is a cyclic group or Wilson's theorem. The proof that $\Bbb R^1\not\cong\Bbb R^n$ for $n>1$ is easy and uses only that the image of a connected set is connected, however that method doesn't generalize nicely. Compare with the homology proof, and we can easily demonstrate $\Bbb R^n\not\cong\Bbb R^m$ (as topological spaces) for any $n\ne m$. The idea of "uniform distribution modulo $1$" gave way to topological group dynamics, whose modern techniques have proven very powerful indeed.
In short there are some techniques that are only able to go so far, it often takes a genuinely new idea to allow for a great surge forward in the theory, and in this spirit the elementary proof presented an opportunity for such advancement. If it proved to exist, as Hardy noted it might give us greater insight in how to understand the theorem.
It is of course true that we do not "need" that proof for the purposes of establishing the PNT, but that's too simplistic a way to think about the elementary proof, or indeed of any proof which uses different approaches from the original proofs. New proofs have almost inherent merit if they are qualitatively different, in that they encourage use to look in those directions for proving other results which might not yield to the standard techniques to which we are accustomed.
A little bit of superstition
Historically new discoveries in all of the sciences, mathematics included, have been intertwined with politics and superstition. In the ancient days the Pythagoreans had an entire cult dedicated to numbers, which--to the Greeks--meant rational numbers. Indeed, there is some historical evidence to suggest the man who discovered irrational numbers may have been killed over the fact! Other notable examples are complex numbers, which were shunned or ignored for the longest time.
The biggest example I can think of is the axiom of choice. Once upon a time it was of a much more central focus in mathematics. Proofs using it were sometimes rejected by some sectors for not being constructivistic. Brouwer even renounced his own fixed-point theorem proof for not being constructive. I imagine it must just not have sat well with a greater percentage of the mathematical population than it does today (I still know of a good handful who will argue vehemently on the subject). I think especially after the long time where we had a lot less precise formulations of theorems, and less air-tight proofs than we do today, this was a more valid concern, but experience seems to show we really shouldn't worry too much about such things.
That is all to say that the relative importance attached to an elementary proof seems historically similar to desires for constructive proofs of things like fixed points, and to an extent I acknowledge they are useful to have for the purposes of illustration, and sometimes in practice where such things are useful objects. Cantor's work proves there are a lot of transcendental numbers, but Liouville, Lindemann, Baker, et al are those that give us our best examples. At the same time, the community's interest moves on to bigger and ultimately more important things.
I think Hardy was much like any other mathematician of his time: still not too far away from the original proof to wonder about a simpler (or at least more elementary) proof of the PNT, an interest which has greatly faded away with all the many proofs we have today. Even today there is a bit of mysticism with our subject, some things that feel like they ought to be true, even if we cannot prove them. I think Hardy was--mistakenly--in the camp that thought somehow an elementary proof would reveal some deep secret about primes in the way it was proved. It could certainly have seemed that way based on what was known and believed in his time, so it was not unreasonable, it just happened to be incorrect.
In despite of what some may think, a lot of what is studied comes from where the general interest lies--Kronecker vs Dedekind turned a lot if those in the latter's school to push down those in the former's, though that effect has naturally lessened over time. The personalities of the day played a large role in what was deemed "important" to discover. Hilbert's famous problems probably gave direction to thousands upon thousands of careers. Ultimately the field continues to evolve, and things like the elementary proof do have their place in the history and in the practice, it just depends on your personal views on whether "requiring complex analysis" really makes a result "fundamentally deeper" than one that does not.