I believe it doesn't but it could be I am wrong.
If we have a sequence $ \{\frac{1}{n}\}_n$ then the archimedian property plays a central role in proving the limit. For instance, let $\epsilon$ be an arbitary real number, then by a well known corollary of the archimedian property, we have that there exists an $N_{\epsilon}$ such that:
$$ \frac{1}{\epsilon} < N_{\epsilon}$$
We then claim that this is the $ N_{\epsilon}$ required to satisfy the epsilon criteria. I also notice that pretty much every proof of epsilon criteria I've done required some sort of use of archimedean property.
However, I think this is completly unnecessary for epsilon delta criterion. For example let us prove that $x^2$ is continous at any point a $x=a$ in the domain. We can let, $$\delta = \min\left(1, \dfrac{\epsilon}{2|x_0| + 1}\right)$$
In the answer explaining it, it can be seen that no mention of archimedian property is there. The construction seems to be well defined as a consequence of the field structure of $\mathbb{R}$.
If I am indeed correct, then would it be a correct inference to make that epsilon delta proofs in some non trivial sense require less to prove than the epsilon criteria?