DTFT and a Downsampled Sinc Function
You got the term $X \left( e^{j (\omega - 2 \pi)/2} \right)$ wrong. It is centered at $2\pi$ and it is non-zero in the interval $(2\pi-2\omega_0,2\pi+2\omega_0)$. So the results obtained in the frequency domain and in the time domain, respectively, are identical.
Note that the term $X \left( e^{j \omega/2} \right)$ is $4\pi$-periodic, so the term $X \left( e^{j (\omega - 2 \pi)/2} \right)$ makes sure that the spectrum of the downsampled signal is $2\pi$-periodic. Clearly, there is no aliasing as long as $\omega_0&lt;\pi/2$ is satsified.