The following text is taken from Analysis II by Amann and Escher, p. 61:
We denote the sequence of prime numbers by $(p_k)$ [...]. It therefore follows for every $m\in\mathbb{N}$ that $$\prod_{k=1}^m\frac{1}{1-p_k^{-s}}=\prod_{k=1}^m \sum_{j=0}^\infty \frac{1}{p_k^{js}}=\sum '\frac{1}{n^s}$$ where, after “multiplying out”, the series contains all numbers of the form $\frac{1}{n^s}$, whose prime factor decomposition $n=$[...] has no other prime numbers from$p_1,\ldots ,p_m$. Therefore $\sum '(1/n^s)$ indeed contains all numbers $n\in\mathbb{N}$ with $n\le p_m$. The absolute convergence of the series $\sum_n (1/n^s)$ then implies $$\left|\zeta (s)-\prod_{k=1}^m \frac{1}{1-p_k^{-s}}\right|=\left|\sum_{n=1}^\infty \frac{1}{n^s}-\sum '\frac{1}{n^s}\right|\le \sum_{n\gt p_m}\frac{1}{n^{\operatorname{Re}s}}.$$ From [...] (a theorem of Euclid), it follows that $p_m\to\infty$ for $m\to\infty$. Therefore, from [the fact that $\sum 1/n^s$ converges absolutely...], the remaining series $\left(\sum_{n\gt p_m} (1/n^{\operatorname{Re}s})\right)$ is a null sequence.
And this is the supposed end of the proof of Euler's product formula for the zeta function.
Question
This question is about supposed circularity in proving that the number of primes is infinite by Euler's product formula. The conclusion there is that no circularity is involved. But I was looking for a rigorous proof of Euler's product formula and, as it seems, it uses infitude of primes (see "it follows that $p_m\to\infty$ for $m\to\infty$ "). Or am I missing something?
Edit: The proof number 3 here (Euler Product Representation) uses the geometric series argument (as Amann does), yet it doesn't seem to use the infitude primes. Is the infitude of primes in Amann's proof superfluous? I'm really confused.
I been thinking about this a lot and the question put is both subtle and non-trivial. As you have indicated, mathematicians in the published literature are coming to different conclusions on this question which is disconcerting.
The provisional conclusion I've come to (at least in general philosophical/logical terms) is this.
I think that mathematicians are logically obliged to prove that there is a fundamental core consistency to a proposed theory of numbers based on on an agreed set of definitions, axioms and laws and fundamental theorems on primes etc., before utilising a proof by contradiction that effectively begs the question (in this case on the existence of the said core consistency to the theory of numbers before it is actually proved/demonstrated).
Begging the question is a form of circular reasoning, which in some cases can lead to logical fallacies.
This answer goes against the majority of answers to the question you linked to here.