Otherwise is there any reason why the (wo)man trusted to fly the thing is not trusted to correctly hear a pushback clearence and relay same to the groundcrew?
Yes, they keep getting it wrong. Why introduce a layer of communication where one is not necessary?
It does indeed have proven results. Further, whilst the captain remains the commander of the aircraft during push back he's not the one actually in control of the aircraft. He's also not the one who may lose his license for one mistake on the ground (the safety culture of handling agents tends to lag a little behind that of ATC and airlines).