This is an elaboration on my comment below John's answer. Its goal is to give a very brief summary of the history underlying the question; I hope that it is more or less correct.
I think that it is fair to say that from the beginning it was understood that non-vanishing on the line $\Re(s) = 1$ was the main requirement for proving the prime number theorem.
However, in Hadamard and de la Vallee Poussin's approaches, this non-vanishing was fed into an explicit formula (which gives a Fourier-type expansion of the prime counting function, or some variant), which was in turn obtained from the $\zeta$-function by various Mellin transform games. In particular, if I understand correctly, establishing the explicit formula involves pushing the
contour of integration to the left of $s = 1$ (since it involves a sum over the zeroes
of the zeta function), and so doesn't just involve analysis in the
region $\Re(s) \geq 1$.
As noted in anon's answer, Landau formulated an approach to PNT via a Tauberian theorem involving just analysis on the region $\Re(s) \geq 1,$ but as well as the crucial condition
of there being no zeroes on the line $\Re(s) = 1$, there was a growth condition.
In his paper, Wiener discusses earlier Tauberian approaches, including Landau's, and then
refers to Ikehara's theorem (proved in Ikehara's thesis, I believe, under Wiener's supervision) as being the "true theorem" (I'm fairly confident that I'm remembering his language correctly here), i.e. the one with the correct condition, those conditions being simply that there are no zeroes on the line $\Re(s) = 1$.