A question about the proof of Seifert - van Kampen

I think the flaw in your reasoning comes earlier in the proof. In the previous paragraph, Hatcher defines two moves that can be performed on a factorization of $[f]$. The second move is

Regard the term $[f_i]\in\pi_1(A_\alpha)$ as lying in the group $\pi_1(A_\beta)$ rather than $\pi_1(A_\alpha)$ if $f_i$ is a loop in $A_\alpha\cap A_\beta$.

Regarding this move, Hatcher asserts that

[This move] does not change the image of this element in the quotient group $Q=\ast_\alpha\, \pi_1(A_\alpha)/N$, by the definition of $N$

This is the step at which Hatcher is using the hypothesis that $N$ is normal. In particular, if $N$ were simply the subgroup generated by the elements $i_{\alpha\beta}(\omega)i_{\beta\alpha}(\omega)^{-1}$ (instead of the normal subgroup generated by these elements), this move would not necessarily preserve the image of this element in $G/N$.

I think it should be pointed out that another style of proof is available, and is used to give a proof of the version using many base points, at this groupoids. With the result for many base points, you can compute the fundamental group of the circle, which is, after all, THE basic example in algebraic topology. The proof given there does only the union of 2 open sets, but it gives the proof by

verification of the universal property,

which is a general procedure of great use in mathematics. For example this method is used to prove higher dimensional versions of the van Kampen Theorem. This method also avoids description of the result by generators and relations.