Why would carbon handlebars be particularly stiff in comparison to alloy bars?
There's no need for them to be either stiffer or less stiff; it depends which carbon bars and which aluminium bars you choose to compare. At the limit, you can make a carbon bar slightly less stiff while remaining reliable, because aluminium ones will fatigue very fast if they are designed to operate at high stress, but typically carbon bars will be stiffer than aluminium bars of the same outline and weight.
There's no need for them to be either stiffer or less stiff; it depends which carbon bars and which aluminium bars you choose to compare. At the limit, you can make a carbon bar slightly less stiff while remaining reliable, because aluminium ones will fatigue very fast if they are designed to operate at high stress, but typically carbon bars will be stiffer than aluminium bars of the same outline and weight.