What would be the point of that? Wars and support of wars do not generally rely on public support. For instance here in the us, only around 3% of americans vote based on foreign policy. Does it really matter which narrative the masses believe? I would think it would be people in power worth persuading, and there are much more direct ways of buying politicians and career government workers.
Propagandizing their own people I get, but what you're outlining just doesn't make sense. "Spreading chaos" does because it draws resources away from their interests to domestic discord.