If jet engines are smaller in diameter than propellers, is it fair to say that propellers accelerate a lot of air a little while a jet engine accelerates a little air a lot? Basically, a jet engine accelerates the air flowing through it by more than a propeller engine of the same thrust, right?
If so, propellers should be way more efficient because, by accelerating a larger mass of air, they're able to generate more force (thrust) while performing less work (change in kinetic energy of the accelerated air) due to the fact that force is proportional to $\Delta V$ but energy is proportional to $\Delta V^2$, but both are proportional to mass.
So, if this is true, shouldn't propulsion systems try to maximize the mass of air that it accelerates in order to minimize the $\Delta V$ required? This is, after all, the whole concept behind using large wings to stay aloft (they can grab onto a large mass of air).
So how can jet engines be more efficient than propeller engines? What other factors contribute to the efficiency of the propulsion system besides just the mass and velocity of accelerated air? And similarly, why aren't propellers way larger than they typically are? Because by this logic airplanes should have propellers the size of helicopter rotors.