If $(X,\mathcal{S})$ is standard Borel space and $\mu$ a continuous measure on $(X,\mathcal{S})$ then there is a Borel isomorphism $F:X\to [0,1]$ that sends $\mu$ to Lebesgue measure on $[0,1]$. (See Kechris 17.41.) Since the isomorphism preserves measure this shows that any measurable subset of $[0,1]$ has measurable preimage under $F$. In other words, if all sets of reals are measurable then so are all subsets of $X$. So one does not have to look at the specifics of Solovay's model, nor at Haar measure for particular groups, as long as one restricts attention to Polish measure spaces.
If $(X,\mathcal{S})$ is standard Borel space and $\mu$ a continuous measure on $(X,\mathcal{S})$ then there is a Borel isomorphism $F:X\to [0,1]$ that sends $\mu$ to Lebesgue measure on $[0,1]$. (See Kechris 17.41.) Since the isomorphism preserves measure this shows that any measurable subset of $[0,1]$ has measurable preimage under $F$. In other words, if all sets of reals are measurable then so are all subsets of $X$. So one does not have to look at the specifics of Solovay's model, nor at Haar measure for particular groups, as long as one restricts attention to Polish measure spaces.