Is every subgroup of a connected unimodular (matrix) Lie group also unimodular?

Take the so-called ax+b group, i.e. the connected component of the affine group of the real line. Or, even more concretely, $$\left\{ \left( \matrix{ a & b \\ 0 & 1 } \right) \colon a>0, b\in{\mathbb R} \right\}.$$ This is not unimodular.

So I think the "proof by handwaving'' has a mistake somewhere. Probably your conception of restricting a measure from a space to a closed subset may need rethinking ...