"Why of all functions does one have to put the Gamma-function there?"
zeta(s)$\zeta(s)$ has trivial zeroes at -2, -4, -6$-2, -4, -6$, etc. zeta(1-s)$\zeta(1-s)$ thus has trivial zeroes at s=3, 5, 7$s=3, 5, 7$, etc - a completely different set of zeroes.
To make a reflection formula where zeta(s)$\zeta(s)$ is somehow equal zeta(1-s)$\zeta(1-s)$, you have to get rid of the two differing sets of trivial zeroes. Multiplying by the gamma is perfect for this since its poles will cancel out those zeroes. For example, gamma(s/2)$\Gamma(s/2)$ has poles at 0, 2, 4, 6$0, 2, 4, 6$, etc. and should go with zeta(s)$\zeta(s)$. gamma((1-s)/2)$\Gamma((1-s)/2)$ has poles at s=1, 3, 5$s=1, 3, 5$, etc. and should go with zeta(1-s)$\zeta(1-s)$.
It's possible to prove that gamma is the right choice, but Euler no doubt discovered that gamma is the right function through numerical experimentation - when he discovered the zeta reflection formula like 250 years ago.
