Why these 'elementary' facts do not solve the Inverse Galois Problem?
If $H\subset \mathrm{Gal}(K/\mathbb Q)$ then the set elements elements of $K$ fixed by $H$ is a field $k$, and you get $$H\cong\mathrm{Gal}(K/k)\,.$$
But the inverse Galois question is seeking $k$ so that $H\cong \mathrm{Gal}(k/\mathbb Q)$.