Explanation relevant to the filter response ($\mathrm{sinc}$ interpolation) - Equation $(2)$ in my question
Eq$(1)$ doesn't look right. It should be:

$$x'(t) = \sum_{n=-\infty}^{+\infty} x(nT_s) \mathrm{sinc}(t-nT_s)$$

For your doubt, consider subsampling (lowering $F_s$) as equivalent to reconstructing $x(t)$ and then sampling it at a lower frequency.

Eq$(2)$ comes from the combination of $2$ lowpass filters:


The ideal reconstruction filter (from eq.$1$)
a new anti-alias filter previos to sampling at the lower $F_s$


These $2$ filters combined should result in eq$(2)$ (several lowpass filters in cascade is equivalent to one filter with the minimum bandwidth, and the product of the gains).