In the Elementary proof of the prime number theorem - Need? I address a bit about this, but I agree your question is distinct.
What were the reasons that proofs without complex analysis were preferred?
As I have heard from Jeffrey Vaaler, and others who were around closer to the time in question, there was something of a belief that a proof without analysis would somehow illuminate something deep about the complexity of the results involved. Evading the use of complex analysis would mean that it was perhaps "less difficult than we thought," or at least this was the view at the time. This reflected the desire to understand the exact nature of the theorem, was it fundamentally related to arithmetic or was the excursion into the Fourier side of things truly necessary, and if so: why? This, to me at least, seems quite natural for a mathematician, as we seek not just to know results but to understand them correctly. Lagrange's theorem is really a statement about the homogeneous nature of groups, something that gives motivation and intuition for things like topological groups, even though the former is just a statement about divisibility of orders. There are, of course many more examples, but as you also have the response in the other topic, I'll curtail further exposition on my part for this bullet point.
Is this distinction still important today?
I haven't felt that at all in the conferences, papers, et cetera that I've read in today's number theory environment. It seems that was a popular sentiment at the time, but--especially following the lack of really interesting new results that could be proved with the Erdös-Selberg proof--that has died down significantly to where it is no longer detectable, even among those who were immediate successors of mathematicians like Hardy.
A lot like proofs that don't use the axiom of choice, it's become less of a community preoccupation over time. In the same way that we all somehow "get used to" new results in mathematics and become less wary as the details of the proof get easier to understand with repetition of reading, things like the axiom of choice have become routine. I think this is an exact analogue with the elementary proof, in despite of all the solid foundation, the culture at the time simply valued elementary techniques more highly. As amazingly useful as harmonic analysis is today, it wasn't all the way as developed as it is now. The Wiener-Ikehara theorem was only published in 1931, and Tate's thesis which brought things to a very interesting point with more abstract harmonic analysis rather than strictly on Euclidean space came in 1950, and Rudin's text on Fourier analysis on groups was in 1962, which was after Hardy's time.
In short more modern approaches were developed, tried by fire, and found to be superior to many classical proofs and techniques. In a sort of academic survival of the fittest, the elementary approach was found unfavorable by the large majority after it was unable to deliver what was hoped, and other techniques have since supplanted them as the more "in vogue" ways to approach problems. There are certainly places where excellent new mathematics comes out of elementary approaches, UIUC for one has many excellent number theorists who are skilled at such things. I do not think the number theory community at-large looks with greater favor upon such approaches like it did in Hardy's day.