Another approach is through the Laurent series of the Riemann zeta function at $s=1$,
$\zeta(s)=\frac1{s-1}+\sum_{n=0}^\infty\frac{(-1)^n}{n!}\gamma_n(s-1)^n\;,$
where the $\gamma_n$ are the Stieltjes constants. Multiplying by $s-1$ yields
$(s-1)\zeta(s)=1+\sum_{n=0}^\infty\frac{(-1)^n}{n!}\gamma_n(s-1)^{n+1}\;.$
Theorem 2.2.2 of Entire Functions by Ralph Philip Boas expresses the order $\mu$ of an entire function given by a power series
$f(z)=\sum_{n=0}^\infty a_nz^n$
in terms of the coefficients:
$\mu=\limsup_{n\to\infty}\frac{n\log n}{\log (1/|a_n|)}\;.$
To evaluate the limit superior, we can use bounds found by Matsuoka: For all $n\ge10$,
$|\gamma_n|\le\frac{\exp(n\log\log n)}{10000}\;,$
and for infinitely many n
$|\gamma_n|\gt\exp(n\log\log n-n\epsilon)\;.$
By substituting Stirling's approximation for the factorial,
$\log n!\sim n\log n -n\;,$
we can see that the limit superior is $1$: The upper bound on $\gamma_n$ ensures that the quotient is eventually below $1+\delta$, and the lower bound ensures that it is infinitely often above $1$.