This question is related to how batteries work. In a battery, the electrons can only flow in conductors, because they can't flow in the solution.
my own thoughts on the matter:
I believe it is related to the "energy" of electrons.
"flow" isn't related to "energy" of electrons quite well. As gold and copper are good conductors not because they have low electron negativity, their electrons are quite low in energy in the potential wells of the sea of nucleus. they are good conductors because, within the deep well, they are "free" to flow.
in water, in order to be free, the electron needs to be so energetic, it has to be above the potential well of water molecules to be free in water. In that case, I don't think the electrons will be free either, the H+ in water should capture those high energy electrons and become gas, as H+ has very high electron negativity. However, In this case, I can't explain how does O- do with his extra electron.
I am aware of the fact that, electrons "flows" in snail pace, it is the "pushing" between electrons (like domino) that conducts electricity