From the turn of the 20th century, the thought of an elementary proof of the prime number theorem was the obvious holy grail for elementary methods, but after the work of Selberg and Erdos in the 40s failed to open up the new avenues that some had hoped, it seems that finding elementary proofs for theorems on the distribution of the primes went out of vogue (with the exception of sieve theory, which has recently produced some spectacular results, but which I am not really interested in for purposes of this question).
Furthermore, I haven't been able to locate a good summary of the modern state of the art (modern meaning after the survey article of Harold Diamond from 1982: https://projecteuclid.org/euclid.bams/1183549769). So, I am curious as to what progress has been made, and also as to whether there there any major theorems in analytic number theory that don't currently have a proof by elementary methods?