Another one that can’t tell the difference here, having ridden both alloy and carbon Easton bars. I struggle to believe the “carbon flexes more” thing personally, carbon has substantially the highest modulus of any material used to build bikes (steel is closest), so for components of a fixed size such as handlebars carbon bars will flex less unless they have far thinner walls, which I don’t think they usually do.
Don’t forget steel is the same modulus in all direction where as carbon fibre is only close in the 0deg direction when used fully in UD guise which won’t be used in real life (well it might be but not on a bike). So as soon as you start making a layup the combined modulus migrates down to that of aluminium.
The specific properties are very good yes, and you have the ability to put the fibres not only in directions you want but spaced out from the neutral axis to maximise their effect. Ie if you put all your UD out at the edges it’s a bit like an I beam compared to a a flat plate.
The reason steel is seen to flex more is the high strength means you can use thin sections and use small diameters to get away from geometric effects.
You can easily make an aluminium bike or a carbon bike stiffer than a steel bike for the same weight due to the specific properties and the change in section profile you can adopt with higher specific modulus materials.
My experience comparing Easton EA70 bars and carbon Havoc bars is there is a definite difference in vibration. But this is well known in other sports such as tennis, hockey, baseball etc and things like axe and hammer handles (fibreglass) where composites are partly used for their damping.